University of Southern California Dissertations and Theses

Explorations in semi-classical and quantum gravity

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Explorations in semi-classical and quantum gravity

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Content Explorations in semi-classical and quantum gravity by Felipe Rosso A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS) August 2021 Copyright 2021 Felipe Rosso Para Danu, que me acompa~ n o y siempre apoy o en esta aventura. ii Acknowledgements I would like to start by thanking my advisor, Cliord V. Johnson, who provided invaluable guidance and advice during my PhD studies. I am grateful for his patience and willingness to have lengthy discussions about physics in order to address my (sometimes annoying) questions, and clear up any confusion I had along the way. My hope is to someday have at least half of Cliord's intuitive and deep understanding of the laws of physics. I am also very grateful to Krzysztof Pilch, who taught me that any dierential equation or ugly computation can be solved if you are willing to ercely look at it long enough. The in uence and support Krzysztof provided during these last four years has been immeasurable. All the other members of the high energy group at USC greatly contributed towards my current understanding of physics, as well; specically, Robert Walker, Ashton Lowenstein, Nicholas Warner, Itzhak Bars and Avik Chakraborty. In addition, I learned a lot from taking some amazing classes at USC with Paolo Zanardi and Hubert Saleur. I would especially like to thank the high energy group at the University of Buenos Aires, where I took my rst steps in theoretical high energy physics. I am grateful to David Blanco, Mauricio Leston, Gaston Giribet, Alan Garbarz and Guillem Perez-Nadal, as well as Joan La Madrid and Ignacio Borsa. During my PhD I have been lucky enough to collaborate in projects with some amazing indi- viduals from who I have learned so much: Cliord V. Johnson, Krzysztof Pilch, Nikolay Bobev, Vincent Min, Andrew Svesko, Gaston Giribet and Laura Donnay. This thesis would not exist without the continuous and unconditional support from my family. Ever since I was little, my father Jorge and mother Ana encouraged me to work hard in order to be the best version of myself; a lesson I still carry with me every time I do research. I thank my siblings Julia, Marina, Santiago and Soa who helped keep me grounded and withstood the most annoying brother anyone could have. Finally, I owe everything to Danu. She is the bravest person I know, who (for some reason) decided to support me in this journey, no questions asked. I can only hope to someday be as fearless and big-hearted as her. iii Contents Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii I The Achronal Averaged Null Energy Condition 1 1 Introduction 2 1.1 Classical energy conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Energy conditions in semi-classical gravity . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Achronal ANEC for conformal theories 9 2.1 Exploiting conformal symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.1 Taking the null plane on a conformal journey - Take I . . . . . . . . . . . . . 10 2.1.2 Mapping the Minkowski ANEC . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Monotonicity of relative entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.1 Taking the null plane on a conformal journey - Take II . . . . . . . . . . . . . 22 2.2.2 Modular Hamiltonian of null deformed regions in curve backgrounds . . . . . 26 2.3 Holographic theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.1 General features of bulk AdS with de Sitter boundary . . . . . . . . . . . . . 32 2.3.2 Curve ansatz and no bulk shortcut . . . . . . . . . . . . . . . . . . . . . . . . 34 3 Achronal ANEC for general QFTs 39 3.1 Near extremal horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1.1 Non-conformal free scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.1.2 General QFTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 De Sitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.3 Anti-de Sitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4 An application: Quantum BMS transformations 63 4.1 Asymptotic (conformal) Killing vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 68 iv 4.1.1 Conformally at space-times . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.1.2 Three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.1.3 Arbitrary dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2 CFT charges and algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.2.1 Conformal transformation of the charges . . . . . . . . . . . . . . . . . . . . . 78 4.2.2 Discrete transformation between future and past regions . . . . . . . . . . . . 82 4.2.3 Charge algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.3 Quantum transformations on the Hilbert space . . . . . . . . . . . . . . . . . . . . . 86 4.3.1 Group action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.3.2 Algebra representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.4 Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.4.1 Three dimensional boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.4.2 Arbitrary dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.5 BMS and black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 II Matrix models and quantum gravity 113 5 Introduction 114 5.1 Topological expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.2 Tesselating surfaces and random matrices . . . . . . . . . . . . . . . . . . . . . . . . 116 6 Random matrix models 123 6.1 Loop equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.1.1 Spectral curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.1.2 Recursion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.1.3 Double scaling limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 6.2 Method of orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.2.1 Finite N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.2.2 Double scaling limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7 Two dimensional quantum gravity 150 7.1 Jackiw-Teitelboim gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 7.1.1 Disc partition function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 7.1.2 Topological expansion and random matrices . . . . . . . . . . . . . . . . . . . 159 7.2 N = 1 Jackiw-Teitelboim supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . 162 7.2.1 Topological expansion and random matrices . . . . . . . . . . . . . . . . . . . 165 7.2.2 Non-perturbative denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 A Appendix 183 A.1 Symmetry constraints on the vacuum stress tensor . . . . . . . . . . . . . . . . . . . 183 A.2 Light-ray operator algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 v A.3 Discrete symmetry of quantum theory . . . . . . . . . . . . . . . . . . . . . . . . . . 189 A.4 Commutator identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 A.5 Bulk superrotation integral curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Bibliography 194 vi List of Tables 1.1 Summary of the most common classical energy conditions, wheret i is a unit normal future pointing timelike vector andk is null. The Weak Energy Condition is implied by the Dominant by settingt 1 =t 2 =t . The Null Energy Condition is the weakest, as it is implied by all the others. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1 Details of the conformal transformations relating the Lorentzian cylinder to various space-times. We indicate the new coordinates, the conformal factor w 2 obtained from the change of coordinates ds 2 cylinder =w 2 d s 2 and the metric of the transformed space-time. The null coordinates in the cylinder are =, while in Minkowski we dene r =rt with the radius r 0 and t2R. . . . . . . . . . . . . . . . . . . 11 2.2 Conformal factor w(;~ x ? ) 2 relating Minkowski and several other curved manifolds, evaluated along the mapped surface obtained from the null plane. In the second row we indicate the ane parameter in each case, where forRS d1 and AdS 2 S d2 it is not given by but 2 [=2;=2] instead. This determines whether the mapped geodesic is complete or not. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Summary of the mapping of the Minkowski null plane under the conformal trans- formations discussed in this section. We indicate the conformal factor w 2 (;~ x ? ) relating the Minkowski null plane to each of the space-times, as well as the ane pa- rameter in the null plane and the one in the mapped surface. The last two columns indicate the induced metric and whether the surface ts in the mapped space-time (see gure 2.5). The metric on the unit sphere S d2 in stereographic coordinates is given by d 2 (~ x ? ) =d~ x ? :d~ x ? =p 2 (~ x ? ), where p(~ x ? ) = (1 +j~ x ? j 2 )=2. . . . . . . . . . . 26 4.1 Summary of our proposal for the holographic description of the boundary states (4.4), that correspond toj p i on the rst column withp =T;R;D respectively. The boundary charge b Q in the rst column are written in (4.3), where we add a hat to remind ourselves it is an operator. The metric g ( p ) in the second column is obtained by acting on the pure AdS d+1 metric with the (nite) transformation generated by p . Q g (p) [ q ] corresponds to the Noether charge associated to the vector q computed in the metric g ( p ). . . . . . . . . . . . . . . . . . . . . . . . 67 4.2 Dierent choices for the conformal factor w 2 (x ) in (4.9) that result in a variety of interesting space-times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 vii 4.3 Non vanishing ordinary and conformal Lie derivatives of the metric g and g = g =w 2 (x ) when considering the vectors in (4.17) for arbitrary functionsf(),Y (), h() and g(). Since the vectors T (f) and R (Y ) have vanishing divergence we can replace the ordinary Lie derivatives in the rst two columns by the conformal derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.4 Non vanishing ordinary and conformal Lie derivatives of the metric g and g = g =w 2 (x ) when considering the vectors in (4.24) for arbitrary functions f(~ y ) and Y A (~ y ), while h(~ y ) and g(~ y ) are restricted to (4.27). . . . . . . . . . . . . . . . . . . 76 4.5 Action of the charges (4.31) on the vacuum statej0i and the statejYi =e iR(Y ) j0i. The rst three columns correspond to the charges (4.57) that generate ordinary conformal transformations. By \6= 0" we mean the state is not an eigenstate of the charge, but it has a non-trivial action. . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.6 Summary of our holographic proposal relating the boundary states (4.105) to the bulk metrics (4.106), where p =T;R;D. For the boundary charge b Q on the rst column we add a hat to remind ourselves it is an operator, written in terms of the stress tensor as in (4.30). The metricg ( p ) in the second column is given in (4.106), obtained by acting on the pure AdS d+1 metric with the (nite) transformation generated by p . Q g (p) [ q ] corresponds to the Noether charge associated to the vector q computed in the metric g ( p ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.7 Action of the vectors p withp =T;R;D on the bulk metric, where all the quantities in this table are evaluated at = 0. These results should be compared with the boundary CFT computations in the rst six columns of table 4.5, where we nd perfect agreement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.8 Gravitational Noether charges of the vectors p withp =T;R;D in (4.101) computed in a pure AdS background and the perturbed metrics in (4.117). The divergences in some of the charges arises when taking the boundary limit ! 0. Comparing with the boundary CFT results, given in the last three columns of table 4.5, we nd perfect agreement when considering quantities that do not involve the superdilation vector D (g). We add a box on the entries where there is disagreement. . . . . . . . 105 6.1 Classication of the dierent types of potentials in a Hermitian random matrix model. The critical potentials are characterized by the rate at which 0 () van- ishes at c , withk2N. In the last column we indicate whether the system is stable or not. The highlighted row corresponds to the critical models that are the focus of this section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.2 Ansatz for the dierent parameters in the matrix model involving the double scaling of a critical model with potential V 2k () (6.21) perturbed by other critical models with i < k (see (6.97) for the full potential). The perturbation away from the k model is parametrized by the coecientsft 2i g k1 i=1 . The double scaling limit ! 0 of the string equationS k (6.98) can be worked out using the rst four rows of this table. The normalization coecient c 2k is xed so that when double scaling a single critical model we have K 2k = r(x) 2k+1 +O(~ 2 ). For the rst few values we have c 2k = (1; 3=8; 1=8; 5=128). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 viii List of Figures 1.1 Sketch of the setup needed to show the classical energy conditions are violated by all axiomatic QFTs dened in Minkowski. The operatorO V is supported on an open set (red) in the causal domain of V (green). For the particular case in whichO V =T the operator is local, i.e. supported on a single point. . . . . . . . . . . . . . . . . . . 6 2.1 Diagrams illustrating the eect of the conformal transformations given in table 2.1 when applied to the Lorentzian cylinder. The whole innite strip on the left diagram corresponds to the cylinder, with the North and South pole at = 0 and = respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Geodesics in the Lorentzian cylinder and de Sitter for several values of ~ x ? . The diagram in the left does not contain all the information since it is missing the motion in the coordinates ~ v on the S d2 . In the center diagram, we plot some trajectories for the case d = 3 where the spatial section of the cylinder is given by S 2 . Equal colors in each diagram correspond to the same geodesics. To the right we have the geodesics in the (;) plane together with the region covered by de Sitter. Since the topology of dS is the same as the cylinder, the trajectories in dS are also given by the center diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 In the left diagram we plot the AdS geodesics in the (;) plane. Comparing with gure 2.2 we see that only half of the curves withj~ x ? j< 1 t inside the space-time. In the right diagram we plot the trajectories in a cross section of the solid cylinder for the case of d = 3. Equal colors in each diagram correspond to the same geodesics. 16 2.4 Plot ofDA + (green) andDA (red) in the (T;X) plane in Minkowski space-time. The function A(~ x ? ) (blue dot) changes along the null plane XT = 0 as we move through the transverse directions ~ x ? outside of the page. A very nice 3D picture of this region can be found in gure 1 of [1]. . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5 Mapping of the null plane in Minkowski to the null surfaces in the (;) plane. The dierent diagrams indicate the sections of the cylinder covered by each of the space- times. We see that discontinuity at =p(~ x ? ) in the Minkowski case is nothing more than the surface going from the Minkowski patchM n=0 toM m=0 in the second diagram. For de Sitter we consider a slight variation of the Weyl rescaling so that the surface ts in the space-time, see discussion above (2.40). . . . . . . . . . . . . . 24 2.6 Plot of the mapped surfaces A and their causal domainsD A in the (;), where we indicate the region of the cylinder covered by each of the space-times. . . . . . . 28 ix 2.7 On the left we have a diagram of the setup in the (;) plane, with the shaded blue region corresponding to de Sitter. The boundary curve is shown in blue, while in green and red we show to dierent bulk curves going out of the page which satisfy the boundary conditions (2.70). The no bulk shortcut property implies that if these bulk curves are causal, the red trajectory is forbidden. On the right we plot our ansatz for the function f z ( ) which shows how the bulk curve goes into the bulk for dierent values of L. As ! 0 the depth of the curve decreases and since 0 = d1 , its range in the boundary goes to 2 [0; 2]. . . . . . . . . . . . . . . . . . . . . . . . 35 3.1 The left diagram corresponds to the Penrose diagram of the exterior region of the extremal black hole in the coordinates (=r h ;) (3.3). The Minkowski null innity I is indicated in green, while the future and past horizon of the black holeH in blue. Several xed r trajectories are plotted in gray. The right diagram gives the maximally extended space-time, with the time-like singularity in red. . . . . . . . . . 40 3.2 Penrose diagram of the near horizon limit of the exterior region of an extremal black hole, which corresponds to the Poincare patch of AdS 2 . The boundaries are located at = 0; while the black horizonH at =. We have plotted several constantr curves in gray. To the right we have the maximal extension, corresponding to global AdS 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3 On the left diagram we sketch the regionD(A 0 ) described in (3.18). The deformed regionD(A) shown in the right diagram contains the additional green section of size (~ v ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.4 On the left diagram we have the Euclidean ow generated by (3.26) applied on the surface at E = 0 + for 2 (0; 2). When we analytically continue to Lorentzian time E =i we obtain the ow on the right diagram. . . . . . . . . . . . . . . . . . 48 3.5 Sketch of the boundary @M E obtained from the regions R (in blue) and C (in green) described in (3.41) as;b! 0. The red wiggly line corresponds to the branch cut in the Euclidean space-time located at E = 0 and 2 (0;=2). . . . . . . . . . . 52 3.6 Plot of the two kernels in the integral in (3.47) for several values ofb. The left (right) diagram corresponds to the integral with positive (negative) s. . . . . . . . . . . . . 54 3.7 Constant time section of AdS 3 given by the regionx 2 +y 2 < 1. We have plotted the constant and trajectories obtained from (3.64). The half space A 0 dened from x> 0 corresponds to <=2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.1 Penrose diagrams associated to AdS 2 S d2 , de Sitter and the Lorentzian cylinder RS d1 . The coordinates (u;) only cover the shaded blue region in each case, where several constant trajectories are sketched in gray. Future null innity at = 0 in Minkowski is mapped to the future horizon H + indicated with a dashed green line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2 On the left we have the region of AdS 2 covered by the coordinates (v;%) dened in (4.46). The past horizon (marked in red) is obtained as % = 0 and v2R. On the right we plot the region of AdS 2 covered by the coordinates (u;) and (v;%) in green and red respectively. The CRT transformation in (4.52) gives a map between these regions, most importantly mapping between the horizons H + $H . . . . . . . . . . 83 x 4.3 Diagram representing the region of global AdS 3 (solid cylinder) covered by the Poincar e patch (solid blue region) in the coordinates in (4.97). While the future boundary of the bulk Poincar e patch is given by ! 0, the future null innity of the AdS boundary (the boundary of the boundary) is located at ; ! 0. . . . . . . 99 4.4 On the left we see the maximally extended Penrose diagram of the extremal Reissner- Nordstrom black hole. Taking the near horizon limit we obtain the AdS 2 S d2 space-time on the right, see [2]. The gray shaded region corresponds to the section of the black hole that is well approximated by AdS 2 S d2 . . . . . . . . . . . . . . . 109 4.5 Diagram showing the transformations that map between the four boundaries of the extremal black hole (4.121): I and H . The mapping of the charges under the conformal and CRT transformations where explored on the rst two subsections of section 4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.6 On the left we see the maximally extended Penrose diagram of the near-extremal Reissner-Nordstrom black hole. Notice that in this case the future (past) horizon H + (H ) is actually the past (future) horizon for the asymptotic region on the right. Taking the near horizon limit we obtain the AdS 2 S d2 space-time on the right, see [2]. The gray shaded region corresponds to the section of the black hole that is well approximated by AdS 2 S d2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.1 Three dierent tesselations of a unit sphere S 2 , built entirely in terms of triangles. As the tesselation is rened we observe how it is able to more accurately approximate the smooth surface of the dierentiable manifold. In the center diagram we indicate the orientation of some of the triangles. . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.2 The three ribbon diagrams that contribute toF(;N) in (5.9) to linear order in . The rst two diagrams can be drawn on the plane (or a sphere) and contain two additional closed loops, indicated in red and green. The third diagram can be drawn on a torus and does not contain any additional loop. Each of these give the three terms in (5.11). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.3 Three ribbon diagrams that contribute toh N TrM 4 3 i, where the number of closed loops is indicated with dierent colors. Each of these diagrams contribute with N 2 , N 0 and N 2 respectively, and can be drawn on a sphere, torus and double torus. Below each diagram we indicate the contraction of the external legs in (5.13) required in each case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.1 On the left we plot the large N spectral density (6.24) in (6.25) for 4 > 0 and 2 p 4 =3< 2 < 0. The dashed gray line corresponds to the corresponding poten- tial V (z) = 2 z 2 + 4 z 4 =3. On the right we plot the contours in the complex plane z 0 used to obtain the recursion relations from the loop equations. For z 0 2 (a ;a + ) we observe the branch cut in the spectral curve y(z 0 ). . . . . . . . . . . . . . . . . . 128 6.2 Spectral density 0 (; ) for the potential in (6.78) (dashed curve). As goes to one we approach criticality and there is a phase transition from a double to a single-cut model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.3 Solution R n to the discrete string equationS k=1 in (6.81) for = 1 and N = 14. The red dashed line corresponds to the leading solution given in (6.83). . . . . . . . 142 xi 7.1 The left diagram corresponds to the Penrose diagram of Lorentzian AdS 2 , with the shaded region corresponding to the region covered by the (t;r) coordinates in (7.14). On the right, we have Euclidean AdS 2 (the Poincare disc) as described by the coordinates (;r) in (7.21). The boundary curve x (u) (7.22) determines the location of the boundary @M where the boundary Dirichlet boundary conditions (7.16) are specied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.2 Topological expansion developed in [3] for the Euclidean partition function of JT gravity with a single boundary of renormalized length indicated in green. . . . . . 159 7.3 Spin structures on the circle and torus, where in each case we indicate whether it is even or odd. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 7.4 Plot of the leading solution r 0 (x) obtained from (7.72) after xing t 2k according to (7.75). In the region x < 0 there are two solutions to the implicit constraint, indicated in red (solid) and blue (dashed). We chose the red branch, which satises the boundary condition (6.96) required when solving the string equation . . . . . . . 171 7.5 Plot of the spectral density of the toy model with (c;) = (0; 1) in (7.98), that is relevant for JT supergravity. While the constant black curve (dashed) corresponds to the perturbative result, in blue we include also non-perturbative eects, which are extremely relevant for small q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 7.6 The solid blue curve corresponds to the full non-perturbative numerical solutionr(x) to the string equation (7.106) in the k max =6 truncation with~=1. The dashed red curve is the leading genus solutionr 0 (x) in (7.72), which is actually the exact solution to all orders in perturbation theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 7.7 In the left diagram we plot each spin contribution s = (7.108) to the spectral density, with the dashed line corresponding to the perturbative answer. In the right diagram we sum both contributions and compare it to the perturbative answer 0 (q) in (7.66) (dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 7.8 In the left diagram we plot each spin contribution s = to the spectral density for = 10. In this regime, we observe good matching with the WKB expression in (7.103) corresponding to the gray curves. In the right diagram we sum both contributions and obtain the expected matching to the perturbative result 0 (q) with = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 7.9 In these diagrams we have isolated the non-perturbative contributions to the spectral density for = 1 and = 10. For large we observe the non-perturbative eects are supressed, as predicted by the WKB approximation in (7.103). . . . . . . . . . . 182 xii Abstract The rst part of this thesis explores the interesting failure of the classical energy conditions when incorporating quantum eects through the semi-classical approximation. Our main focus is on the achronal Averaged Null Energy Condition (ANEC), which provides a natural extension of the classical Null Energy Condition. We give several independent proofs of the achronal ANEC for dierent classes of QFTs dened on certain curved backgrounds, providing concrete evidence that points towards the achronal ANEC as a fundamental feature of semi-classical gravity. As an application, we study its consequences on the innite dimensional BMS algebra that arises on certain black holes horizons and asymptotically at space-times. In the second part of the thesis we study two dimensional theories of quantum gravity, where several of the obstacles that arise when trying to quantize gravity in higher dimensions are absent. Recent progress has shown a class of Jackiw-Teitelboim gravity theories can be quantized explicitly and observables computed in a perturbative topological expansion to arbitrary order. Remarkably, the topological expansion is exactly reproduced by certain ensembles of random matrices. We show how the matrix model description can be used to provide a fully non-perturbative denition of the quantum gravity theory that allows for concrete and explicit calculations of observables beyond perturbation theory. xiii Preface This thesis collects some of the work carried out during my PhD at the University of Southern California, from August 2017 to May 2021. It is based on the following publications: F. Rosso, Global aspects of conformal symmetry and the ANEC in dS and AdS, JHEP 03 (2020) 186, [arXiv:1912.08897] F. Rosso, Achronal averaged null energy condition for extremal horizons and (A)dS, JHEP 07 (2020) 023, [arXiv:2005.06476] L. Donnay, G. Giribet, and F. Rosso, Quantum BMS transformations in conformally at space-times and holography, JHEP 12 (2020) 102, [arXiv:2008.05483] C. V. Johnson, F. Rosso, and A. Svesko, A JT supergravity as a double-cut matrix model, arXiv:2102.02227 Additional work that is not covered in this manuscript is: F. Rosso, T T deformation of random matrices, arXiv:2012.11714 C. V. Johnson and F. Rosso, Solving Puzzles in Deformed JT Gravity: Phase Transitions and Non-Perturbative Eects, JHEP 04 (2021) 030, [arXiv:2011.06026] F. Rosso and A. Svesko, Novel aspects of the extended rst law of entanglement, JHEP 08 (2020), no. 08 008, [arXiv:2003.10462] F. Rosso, Localized thermal states and negative energy, JHEP 10 (2019) 246, [arXiv:1907.07699] N. Bobev, V. S. Min, K. Pilch, and F. Rosso, Mass Deformations of the ABJM Theory: The Holographic Free Energy, JHEP 03 (2019) 130, [arXiv:1812.01026] F. Rosso, Holography of negative energy states, Phys. Rev. D99 (2019), no. 2 026002, [arXiv:1809.04793] xiv C. V. Johnson and F. Rosso, Holographic Heat Engines, Entanglement Entropy, and Renor- malization Group Flow, Class. Quant. Grav. 36 (2019), no. 1 015019, [arXiv:1806.05170] F. Rosso, Holographic heat engines and static black holes: a general eciency formula, Int. J. Mod. Phys. D 28 (2018), no. 02 02, [arXiv:1801.07425] xv Part I The Achronal Averaged Null Energy Condition 1 Chapter 1 Introduction This rst part of the thesis explores the interesting failure of the classical energy conditions when incorporating quantum eects through the semi-classical approximation. Our main focus is on the achronal Averaged Null Energy Condition (ANEC), which provides a natural extension of the classical Null Energy Condition. Several independent proofs of the achronal ANEC are given for dierent classes of QFTs dened on de Sitter, anti-de Sitter and AdS 2 S d2 , providing concrete evidence that points towards the achronal ANEC as a fundamental feature of semi-classical gravity. As an application, we then study its consequences on the innite dimensional Bondi, van der Burg, Metzner and Sachs (BMS) algebra that arises on certain black holes horizons and asymptotically at space-times. In this rst chapter we motivate and provide some background that is necessary to understand our results. 1.1 Classical energy conditions Einstein's gravitational eld equations determine the dynamics of space-time as described by the metric tensorg . The full action coupled to some matter eld =f; a ;A g can be written as 1 I[g ; ] = 1 16 Z d d x p g (R 2) +I M [g ; ] ; (1.1) where is the cosmological constant,R the Ricci scalar and I M the action of the matter elds, covariantly coupled to the space-time metric. Varying with respect to g we obtain Einstein's 1 Throughout this thesis we use Planck units in which G =~ =c =kB = 1. 2 equations 2 R 1 2 g R + g = 8T ; where T = 2 p g I M g : (1.2) While we should also consider the equation of motion for the matter eld (obtained from varying I M [g ; ] with respect to ) it is sometimes useful to study the metric equations by themselves. Given a particular stress tensorT , the space-time metricg is obtained by solving (1.2). Early on, it was realized this logic can be inverted, meaning that for any smooth metricg we can compute the left-hand side of (1.2) and determine the appropriate matter required to produce it. This is problematic, as there is an abundance of mathematically sensible metricsg one can construct that are not \physically" meaningful, e.g. space-times which allow for superluminal trajectories or closed time-like curves [19]. To avoid having these unwanted solutions in the theory, a set of constraints on the stress tensorT , known as Classical Energy Conditions, were introduced [20{22]. See table 1.1 for the most common examples. These conditions are not based on some fundamental principle but are simple constraints that allow for a general analysis of physical solutions to Einstein's equations. Broadly speaking, they are motivated by the intuition that gravity is always supposed to be an attractive force. 3 Arguably the most interesting of these conditions is The Null Energy Condition (NEC) : T k k 0 ; (1.3) where k is a null vector, i.e. k k g = 0. Taking the NEC as a hypothesis, one can prove important results in general relativity, most notably Hawking area [25] and Penrose singularity theorems [26] (see [27] for a review). Although the NEC is satised by many classical matter elds, there are certain theories that violate it. The simplest example is given by a free scalar eld I M [g ;] = 1 2 Z d d x p g g (@ )(@ ) + (m 2 +R) 2 ; (1.4) 2 An alternative way of dening the stress tensor e T involves varying the total action (1.1) with respect to the metric, setting it to zero and identifying e T after rearranging asR 1 2 gR = 8 e T . While the denition ofT in (1.2) agrees with e T for all matter elds that are minimally coupled to the metric when = 0, they can disagree if there is non-minimal coupling. For instance, one gets T 6= e T for the matter action (1.4) with 6= 0. More explicitly, compare (1.5) with e T = ( 0 () 2 @ 2 () 2 )=(1 8 2 ). In this thesis, we always use the denition given in (1.2) and explorations involving e T are left to future work. See [16{18] for a detailed discussion, in particular equations (2.4) and (2.27) of [16]. 3 This is seen more clearly from the Raychaudhuri's equation [23, 24], which determines the congruence of null or time-like curves. For instance, combining the Raychaudhuri and Einstein's equations together with the Null Energy Condition implies non-rotating null congruences locally converge, meaning gravity is an attractive force when moving along null trajectories. 3 Classical Energy Conditions Null Energy Condition T k k 0 Weak Energy Condition T t t 0 Dominant Energy Condition T t 1 t 2 0 Strong Energy Condition T t t + 1 d 2 T g 0 Table 1.1: Summary of the most common classical energy conditions, where t i is a unit normal future pointing timelike vector and k is null. The Weak Energy Condition is implied by the Dominant by setting t 1 =t 2 =t . The Null Energy Condition is the weakest, as it is implied by all the others. where 2 R controls the non-minimal coupling to the Ricci scalar. The stress tensor of this theory can be easily computed from its denition in (1.2). Projecting along the tangent vector k =dx =d of a null curve x () (parametrized by an ane parameter ) we nd T T k k = 0 () 2 +(R @ 2 )() 2 ; (1.5) where() =(x ()) and we have denedA =A k k for any tensorA . 4 Since the second term in this expression is not positive denite, the NEC is only satised for the minimally coupled scalar eld, in which = 0 and T = 0 () 2 0. All the other energy conditions in table 1.1 can also be violated by this scalar eld when 6= 0. 1.2 Energy conditions in semi-classical gravity The status of the energy conditions in table 1.1 becomes even more critical when quantum eects are taken into account. Instead of trying to study a full quantum gravity theory let us consider semi-classical gravity, in which matter degrees of freedom are treated quantum mechanically on a classical space-time. More precisely, the backreaction of quantum matter elds is incorporated through the following modication of Einstein's equations R 1 2 g R + g = 8hT i : (1.6) 4 In deriving this expression we have used the equations of motion and the fact that is an ane parameter, meaning k satises the geodesic equation k rk = 0. 4 The crucial dierence is on the right-hand side, where we now have the expectation value of the operator T of the matter eld on the statej i. 5 While this semi-classical equation is intuitive and seems harmless, several subtle aspects must be addressed when trying to solve it in a particular setup (see [28] for a detailed discussion). In this context, the energy conditions must be considered for a Quantum Field Theory (QFT) on a xed curved background. The positivity of T as an operator becomes even more problematic than in the classical case, essentially due to the fact that quantization and positivity do not mix well. To see this, let us consider the simple case of a free scalar in Minkowski, in which the eld squared is (schematically) given by (x) 2 (a +a y ) 2 , where a y and a are the usual creation and annihilation operators. While (x) 2 is positive, high energy excitations that are present in all states of the Hilbert space make the expectation value of (x) 2 divergent. To obtain a sensible operator this divergence must be regulated by redening the operator through the normal ordering prescription : (x) 2 : a 2 + (a y ) 2 + 2a y a. While : (x) 2 : is a well dene operator, it is not positive anymore, as can be easily shown by taking the expectation value on a state built from a superposition of the vacuum and a two particle state. This means that even for the minimally coupled scalar eld with = 0, the null component of the stress tensor : T : = : 0 () 2 : is not positive denite and violates the NEC. There is actually a very general argument due to Epstein, Glaser and Jae [29] that shows axiomatic quantum eld theories in Minkowski do not admit local and positive operators. Given a Cauchy surface , let us pick the causal domainD(V ) of an open set V , see gure 1.1. From the algebra of operatorsA(V ) localized in the regionD(V ), let us assume there is an operator O V that is positive and has vanishing expectation value on the vacuum stateh0jO V j0i = 0. From these assumptions we immediately concludeO V annihilates the vacuum, given thatk p O V j0ik 2 = h0jO V j0i = 0. The next step is to take the surface V c such that =V[V c , so that the algebra of operators in its causal domainD(V c ) is given byA(V c ). IfD(V c ) is an open set, the Reeh- Schlieder theorem [30] implies any statej i2H in the Hilbert space can be approximated to arbitrary precision by acting withA(V c ) on the vacuum, i.e.j i =O V cj0i. From this we conclude O V j i =O V O V cj0i =O V cO V j0i = 0 ; 8j i2H =) O V 0 ; (1.7) 5 From now on, when writing T we are always going to be referring to the operator in the quantum eld theory. 5 Figure 1.1: Sketch of the setup needed to show the classical energy conditions are violated by all axiomatic QFTs dened in Minkowski. The operatorO V is supported on an open set (red) in the causal domain of V (green). For the particular case in whichO V =T the operator is local, i.e. supported on a single point. where we have used [O V ;O V c] = 0. TakingO V = T shows the NEC (or any of the conditions in table 1.1) cannot be satised by any QFT in Minkowski. 6 While the whole argument is built in Minkowski space-time, it is expected to apply to QFTs dened on any space-time that is globally hyperbolic and real analytic [31{34]. All in all, the failure of the NEC in semi-classical gravity is quite dramatic, since it is violated by all matter elds. Instead of being discouraged by this we should become excited, as it opens a small window into the realm of quantum gravity. The violation of the NEC disqualies Hawking's area theorem in semi-classical gravity and allows black holes to evaporate [35], giving rise to the black hole information paradox [36, 37], without question the most intriguing result in quantum gravity. The hope is that by studying and carefully understanding the mechanisms that violate the NEC we learn valuable lessons about semi-classical gravity. The rst step in this direction is given by searching for a replacement of the NEC which holds in semi-classical gravity. In doing so we must keep in mind the result of [29], which (broadly speaking) implies locality and positivity cannot coexist in QFTs. A quantum generalization of the NEC can retain either locality or positivity, but not both. The achronal Averaged Null Energy Condition (ANEC) is a proposal which sacrices locality in favor of positivity. 7 Building on the previous 6 Although the Hamiltonian operator is positive and non-trivial, it is not in contradiction with the argument given that it is supported on the entire Cauchy slice , meaning V c =;. 7 A dierent non-local bound along null geodesics that is dierent from the achronal ANEC was proposed in [38, 39]. The Quantum Null Energy Condition is a proposal which sacrices positivity in favor of a bound on the 6 work in [28, 46, 47], its most rened version was conjectured by Graham and Olum [48] and can be stated as follows: Achronal Averaged Null Energy Condition: Every complete achronal null geodesic on a self-consistent solution in semi-classical gravity satises Z +1 1 dT 0 ; (1.8) where is an ane parameter and T k k T , with k the geodesic tangent vector. Let us explain and motivate the main features of this proposal. By a self-consistent solution in semi-classical gravity we mean solutions for which the curvature radius of the space-time obtained from (1.6) is much larger than the Planck scale. In particular this addresses certain violations of (1.8) obtained by performing a large Weyl rescaling of the metric [49]. A complete achronal null geodesic is one for which the ane parameter can be extended to all real values and cannot be intersected twice by any time-like path. This condition is included to avoid violations by a Casimir vacuum energy in space-times with compact spatial directions, such as RS d1 . The non-local operator in (1.8) bypasses the argument of [29] explained above, given that the space-like region to a complete and achronal null geodesic in Minkowski is the transverse null plane, which is not an open region of the space-time. Therefore, we cannot apply the Reeh-Schlieder theorem and conclude O V 0 as in (1.7). The achronal ANEC is a useful constraint that can be used to prove powerful theorems, such as topological censorship [50] and other interesting results [51, 52]. Moreover, it can be used to rule out space-times with closed time-like curves [53, 54] (see [48] for a more recent discussion). Even in the simplest setup of a QFT in Minkowski, it has been an extremely useful condition to derive the conformal collider bounds [55] and many other interesting results [6, 56{61]. Although the formulation of the NEC applies locally to every point in the manifold, the achronal ANEC refers only to very special setups, i.e. complete and achronal null geodesics. As shown in section 8.6 of [27], these geodesics are actually very rare and non-generic. The main known examples of space-times which contain such complete and achronal null geodesics fall into two categories: 1. Maximally symmetric space-times, such as Minkowski, anti-de Sitter and de Sitter, in which all complete null geodesics are achronal. local stress tensor, see [40{45]. 7 2. Black hole solutions, which contain achronal and complete null geodesic along their horizon. Let us stress that we are not implying these to be the only examples, they simply correspond to the main known and tractable cases. It is for QFTs dened on these space-times that there is hope we might be able to explicitly prove the achronal ANEC in full generality. For general QFTs in Minkowski this has been recently accomplished in [62{64]. While for curved space-times a wide variety of powerful results have been developed [46, 65{69], a proof for general QFTs in any xed space-time is still missing. The main aim of chapters 2 and 3 is to ll this gap, and provide several proofs of the achronal ANEC for CFTs and general QFTs, dened on a xed de Sitter, anti-de Sitter and AdS 2 S d2 background. This gives crucial evidence that points towards the achronal ANEC as a true statement of semi-classical gravity. On chapter 4 we study quantum BMS transformations and explore some interesting consequences of the ANEC in this context. Several technical calculations are relegated to appendix A. 8 Chapter 2 Achronal ANEC for conformal theories The main aim of this chapter is to prove the achronal ANEC in de Sitter and anti-de Sitter, as well as some novel energy conditions for incomplete geodesics in RS d1 and AdS 2 S d2 . Since the calculations presented here heavily rely on conformal symmetry, they are valid only for CFTs. In section 2.1 we introduce powerful conformal transformations and apply them to the null plane in Minkowski. Using that the ANEC in Minkowski has been proven for general QFTs [62, 63], we map it to several curved space-times and show that for de Sitter and anti-de Sitter the condition we obtain is precisely the achronal ANEC. This is not the case for AdS 2 S d2 and RS d2 , where we obtain a dierent set of conditions on achronal but incomplete null geodesics. We continue in section 2.2, where we consider a dierent approach that is instead based on relative entropy. Using a dierent set of conformal transformations, we map the modular hamiltonian of the Minkowski vacuum associated to a null deformation of the null plane, computed in [70]. Applying the monotonicity property of relative entropy, we obtain some interesting conditions, including the achronal ANEC in de Sitter. This sets the stage for the proof of the achronal ANEC for general QFTs that we present in the following chapter. We nish this chapter with section 2.3, where we provide a proof of the achronal ANEC for holographic conformal theories in de Sitter, dual to Einstein gravity. We follow the approach of [71], where the Minkowski ANEC was derived under the assumption that the gravity dual has good causal properties. 9 AdS Boundary AdS Boundary North Pole South Pole North Pole South Pole de Sitter dS past boundary dS future boundary Figure 2.1: Diagrams illustrating the eect of the conformal transformations given in table 2.1 when applied to the Lorentzian cylinder. The whole innite strip on the left diagram corresponds to the cylinder, with the North and South pole at = 0 and = respectively. 2.1 Exploiting conformal symmetry In this rst section we map the ANEC for QFTs in Minkowski to several curved space-times, in this way obtaining the achronal ANEC for arbitrary CFTs in de Sitter and anti-de Sitter. Subsections 2.1.1 and 2.1.2 develop the mapping of the null geodesics and Hilbert space operators respectively. 2.1.1 Taking the null plane on a conformal journey - Take I Since we are going to be dealing with several dierent curved manifolds, let us start by reviewing their description and main features. It is convenient to start with the metric in the Lorentzian cylinderRS d1 , which can be written as ds 2 =d 2 +d 2 + sin 2 ()d 2 d2 (~ v ) ; (2.1) where 2R is the time coordinate and 2 [0;], with the end points corresponding to the North and South pole of the spatial sphere S d1 of unit radius. In this section we use dimensionless coordinates. The line element d 2 d2 (~ v ) = 4jd~ vj 2 (1 +j~ vj 2 ) 2 ; (2.2) 10 Map to New coordinates Conformal factor w 2 Transformed space-time RR d1 r = tan( =2) (1 +r 2 + )(1 +r 2 )=4 dt 2 +dr 2 +r 2 d 2 (~ v ) dS d cosh(t s ) = 1= cos() cos 2 () dt 2 s + cosh 2 (t s ) d 2 + sin 2 ()d 2 (~ v ) AdS d = tan() cos 2 () ( 2 + 1)d 2 + d 2 2 + 1 + 2 d 2 (~ v ) AdS 2 S d2 % = cot() sin 2 () (% 2 + 1)d 2 + d% 2 % 2 + 1 +d 2 (~ v ) Table 2.1: Details of the conformal transformations relating the Lorentzian cylinder to various space-times. We indicate the new coordinates, the conformal factorw 2 obtained from the change of coordinates ds 2 cylinder =w 2 d s 2 and the metric of the transformed space-time. The null coordinates in the cylinder are =, while in Minkowski we dene r =rt with the radius r 0 and t2R. corresponds to a unit sphere S d2 in stereographic coordinates~ v2R d2 . 1 This cylinder manifold can be represented by an innite strip in the (;) plane, as shown in the leftmost diagram of gure 2.1, where the North and South pole are given by the vertical lines at = 0 and = respectively. Other values of 2 (0;) in this diagram corresponds to a unit sphere S d2 . The space-times we are interested in can be obtained from the Lorentzian cylinder by applying a conformal transformation. Broadly speaking, the transformations are essentially given by dierent ways of cutting the innite strip in the (;) plane. The cutting is implemented by a change of coordinates which puts the metric of the cylinder in the formds 2 cylinder =w 2 d s 2 , followed by a Weyl rescaling which removes the conformal factorw 2 . Eectively, this maps a section of the Lorentzian cylinder to the space-time d s 2 . In this way we can obtain Minkowski, de Sitter, anti-de Sitter and AdS 2 S d2 . 2 The appropriate change of coordinates and conformal factors in each case are indicated in table 2.1. From this it is straightforward to see that each of the transformations cuts the innite strip as shown in gure 2.1. For instance, in the Minkowski case r 2 R translates into 2 [;] together with the implicit constraint 2 [0;]. The space-time AdS 2 S d2 is particularly interesting as this is the geometry obtained from near horizon limit of near extremal black holes, as we review in the following chapter. 1 To obtain the S d2 in terms of the usual angles we describe the vector ~ v2R d2 in spherical coordinates and then parametrize its radius according toj~ vj = tan(=2) with 2 [0;]. 2 Starting from the Lorentzian cylinder, [72] discusses some additional conformal relations. 11 The way in which we have written some of the metrics in table 2.1 is (probably) the most familiar form but not the most convenient to describe null geodesics, which is ultimately what we are going to be interested in. Instead it is more useful to describe all the metrics in terms of the original (;) coordinates. For instance, the (A)dS metrics are given by ds 2 dS = d 2 +d 2 + sin 2 ()d 2 (~ v ) cos 2 () ; ds 2 AdS = d 2 +d 2 + sin 2 ()d 2 (~ v ) cos 2 () ; (2.3) Note that due to the denominators in (2.3) the range of is restricted toj=Rj=2 for dS while 2 [0;=2] in AdS. This implements the cutting of the innite strip as sketched in gure 2.1. Let us now consider the null plane in d-dimensional Minkowski and analyze its transformation properties under these mappings. Taking Cartesian coordinates X = (T;X; ~ Y ) in Minkowski, the null plane X =XT = 0 can be parametrized in terms of (;~ x ? ) as N plane = n X 2RRR d2 : X (;~ x ? ) = (;;~ x ? ) ; (;~ x ? )2RR d2 o : (2.4) For xed~ x ? the curve X () trivially satises the geodesic equation d 2 X d 2 + dX d dX d = 0 ; (2.5) since the connection vanishes in these coordinates. This means is an ane parameter and ~ x ? a label that goes through the dierent geodesics in the null plane. Since the transformation from Minkowski to the cylinder in table 2.1 is given in terms of radial null coordinates r =rt, it is convenient to rst change from the Cartesian spatial coordinates (X; ~ Y ) to spherical. We can do this by dening (r;~ v ) according to r = X 2 +j ~ Yj 2 1=2 ; ~ v = ~ Y X + X 2 +j ~ Yj 2 1=2 : (2.6) Using this together with (2.4) we can write the null plane in spherical coordinates, where the Minkowski metric isds 2 =dt 2 +dr 2 +r 2 d 2 (~ v ). The conformal mapping from Minkowski to any of the previously introduced space-times is applied by writing r = tan( =2) with =, so that the null surface in the coordinates v = ( + ; ;~ v ) becomes v (;~ x ? ) = + (;~ x ? ); (;~ x ? ); ~ x ? + p 2 +j~ x ? j 2 ! ; (;~ x ? )2RR d2 ; (2.7) where (;~ x ? ) = 2 arctan p 2 +j~ x ? j 2 : (2.8) 12 RS d1 dS d AdS d AdS 2 S d2 w 2 (;~ x ? ) 2 + 1 +j~ x ? j 2 2 2 1 +j~ x ? j 2 2 2 1j~ x ? j 2 2 2 2 +j~ x ? j 2 Ane parameter = 1 +j~ x ? j 2 2 tan( ) =j~ x ? j tan( ) Complete Geodesic? No Yes Yes No Table 2.2: Conformal factor w(;~ x ? ) 2 relating Minkowski and several other curved manifolds, evaluated along the mapped surface obtained from the null plane. In the second row we indicate the ane parameter in each case, where for RS d1 and AdS 2 S d2 it is not given by but 2 [=2;=2] instead. This determines whether the mapped geodesic is complete or not. The conformal factor relating each space-time to Minkowski evaluated along the mapped surface can be easily computed, so that we results shown in table 2.2. To understand the mapping of the null plane, let us analyze its behavior for xed values of ~ x ? . The geodesic equation (2.5) is not invariant under the conformal transformations since the connec- tion transforms with an additional term under the Weyl rescaling, and becomes d 2 v d 2 + dv d dv d = d d ln w 2 (;~ x ? ) dv d ; (2.9) where is the connection after the conformal transformation. Although this is also a geodesic equation, the right hand side vanishes only when the conformal factor is independent of . This happens for the mapping to de Sitter and anti-de Sitter, which means the parameter is also ane in the mapped space-times. Although this is not the case forRS d1 and AdS 2 S d2 , we can nd an ane parameter by dening () which cancels the right hand side in (2.9) 00 () = 2 d d ln (w()) 0 () =) () =c 0 Z d w 2 () +c 1 ; (2.10) where c 0 and c 1 are integration constants which can depend on the transverse coordinates ~ x ? . In this way we obtain the ane parameters (given by or in each case) indicated in table 2.2, where we have conveniently xed the integration constants c 0 and c 1 . When is ane, the mapped geodesics are complete, given that takes values on the whole real line. This is not the case when is ane parameter, since in those cases its range is instead given by 2 [=2;=2]. Let us now analyze some important features of the mapped geodesics in each case. 13 Lorentzian cylinder: For any value of ~ x ? all the curves begin and end at the same space-time points, given by (;;j~ vj ) initial = 2 ; 2 ; +1 ; (;;j~ vj ) nal = 2 ; 2 ; 0 : (2.11) Remember that theS d2 in the cylinder metric (2.1) is parametrized in stereographic coordinates~ v, so thatj~ vj equal to zero and innity correspond to antipodal points in the S d2 . This means that both the initial and nal points lie on the equator = =2 of the spatial sphere S d1 , but on opposite sides. As the ane parameter takes values in 2 [=2;=2], the curves travel between these points without intersecting and covering the whole sphere. For some special values of ~ x ? the coordinate description of the trajectory are particularly simple. For instance, the geodesics withj~ x ? j = 1 always stay on the equator ==2, and are parametrized according to Forj~ x ? j = 1 =) v ( ;~ x ? ) = 2 + ; 2 ;~ x ? 1 + cos( ) sin( ) : (2.12) Other simple curves are given byj~ x ? j equal to zero or innity, which corresponds to trajectories that go through the North and South pole of S d1 respectively. Their motion in the~ v coordinate is always constant expect at the pole where it discontinuously changes from zero to innity. For all other values ofj~ x ? j the curves travel along other possible paths in the sphere without intersecting. In the center diagram of gure 2.2 we show some trajectories for the case d = 3, where the spatial section of the cylinder is an S 2 . For higher dimensions we can represent the geodesics in the (;) plane as shown in the left diagram of that gure. Although all these curves are null, they are not necessarily at an angle of =4 since they have a non-trivial motion in the coordinate ~ v. Only for j~ x ? j equal zero and innity the coordinate~ v remains constant and the curves have an angle of =4 in the (;) plane. De Sitter: In the right diagram of gure 2.2 we plot the geodesics in the (;) plane. All curves t exactly inside in the space-time, traveling from the boundary at past innity to future innity. Since the topology of de Sitter is the same as the cylinderRS d1 , with a time dependent radiusS d1 , the trajectories are the same as for the cylinder shown in the center diagram of gure 2.2. The dierence is that the curves in de Sitter cannot be extended beyond their initial and nal points, since they encounter the dS boundaries atjj ==2. 14 North Pole South Pole Cylinder De Sitter North Pole South Pole Figure 2.2: Geodesics in the Lorentzian cylinder and de Sitter for several values of ~ x ? . The diagram in the left does not contain all the information since it is missing the motion in the coordinates ~ v on the S d2 . In the center diagram, we plot some trajectories for the case d = 3 where the spatial section of the cylinder is given byS 2 . Equal colors in each diagram correspond to the same geodesics. To the right we have the geodesics in the (;) plane together with the region covered by de Sitter. Since the topology of dS is the same as the cylinder, the trajectories in dS are also given by the center diagram. Anti-de Sitter: This case is quite dierent, since there are geodesics that lie outside the space- time, as we see in the left diagram of gure 2.3. Only curves withj~ x ? j < 1 lie inside AdS. The critical geodesic that has a vertical path in the (;) plane is given byj~ x ? j = 1 in (2.12), and travels exactly along the AdS boundary. This is in accordance with the vanishing of the conformal factor indicated in table 2.2, which is signaling something important since the conformal transformation is not invertible around that point. For d = 3 we plot the trajectories of the AdS geodesics in a cross section of the solid cylinder, so that we get the right diagram in gure 2.3. Dierent values of ~ x ? follow distinct paths in AdS. This is in contrast to the cylinder and dS where all the geodesics are equivalent up to a rotation of the sphere S d1 . The maximum depth in AdS reached by each geodesic is given at = 0, and can be written in terms of the AdS radial coordinates = tan() in table 2.1 as min = 2j~ x ? j 1j~ x ? j 2 ; j~ x ? j2 [0; 1) : (2.13) The maximum depth corresponds toj~ x ? j = 0 where the geodesic reaches the center of AdS, while forj~ x ? j = 1 the geodesics travel along the AdS boundary and min diverges. 15 AdS boundary Initial point Final point AdS center AdS center AdS boundary Anti-de Sitter Figure 2.3: In the left diagram we plot the AdS geodesics in the (;) plane. Comparing with gure 2.2 we see that only half of the curves withj~ x ? j < 1 t inside the space-time. In the right diagram we plot the trajectories in a cross section of the solid cylinder for the case of d = 3. Equal colors in each diagram correspond to the same geodesics. AdS 2 S d2 : In this case, all the geodesics begin and start at the center of AdS 2 , given by = =2. In between, the trajectories travel between the antipodal points of the transverse S d2 atj~ vj = 0; +1. Similarly as for the cylinder, the geodesics are not complete, since they can be extended beyond their initial and nal points, as is manifest in the restricted range of the ane parameter in table 2.2. The complete trajectory is in general very complicated and can be visualized more easily by plotting in the (;) plane. Doing so, we obtain the same plot as for the cylinder in gure 2.2, with the dierence that = 0; correspond to the boundaries of AdS 2 instead of the North and South poles (see rightmost diagram in gure 2.1). These diagrams are not capturing the non-trivial motion in the S d2 sector. The magnitude of ~ x ? controls how close to the AdS 2 boundary the curves travels, withj~ x ? j = 0;1 reaching the boundary. 2.1.2 Mapping the Minkowski ANEC Now that we understand how to map the Minkowski null plane to these curved space-times us- ing conformal transformation, we want to study the transformation of the ANEC operator in 16 Minkowski, that is given by E(~ x ? ) Z +1 1 dT (;~ x ? ) 0 : (2.14) For arbitrary QFTs in Minkowski, this condition has been rmly established in [62{64]. The integral is over a null geodesic in the null plane (2.4), parametrized by and labeled by ~ x ? . The stress tensor T is projected along this null path according to T = dX d dX d T ; (2.15) whereX =X (;~ x ? ) in (2.4). Let us now study the consequences of the Minkowski ANEC (2.14) when applying the conformal transformations we just discussed. To map the integral operator in (2.14) we need the transformation of the stress tensor. Given the Hilbert spaceH associated to the eld theory in Minkowski, the unitary operator U :H! H implements the mapping to H, the Hilbert space of the transformed CFT. Since T is a quasi- primary operator with spin ` = 2 and scaling dimension = d it transforms under the adjoint action of U as UT U y =jw(v )j 2d @v @X @v @X T S : (2.16) The anomalous term S is proportional to the identity operator and non-vanishing for even d. For d = 2 it can be written in terms of the Schwartzian derivative. Assuming T has vanishing expectation value in the Minkowski vacuumj0i, 3 we can determine the anomalous contribution S as 0 =h0jT j0i =) S =h 0j T j 0ih T i 0 ; (2.17) where we have used that S is proportional to the identity operator. The eect of the anomalous term is to ensure that the mapped stress tensor T vanishes when evaluated in the new vacuum statej 0i. Using this we can write the transformation of the operator T appearing in (2.14) as UT (;~ x ? )U y =jw(;~ x ? )j 2d T (;~ x ? )h T i 0 ; (2.18) where the components of T are now computed from the null surface v (;~ x ? ) in (2.7). In this 3 For our purposes this assumption is not strictly necessary. Although Poincare symmetry of the vacuum only impliesh0jTj0i/ , when projecting the stress tensor along the null direction T this constant factor drops out. See appendix A.1 for a more detailed analysis of the constraints on the vacuum stress tensor from symmetry. 17 way, the mapping of the Minkowski ANEC in (2.14) is in general given by UE(~ x ? )U y = Z +1 1 djw(;~ x ? )j 2d T (;~ x ? )h T i 0 0 : (2.19) This gives a non-trivial positivity constraint for CFTs dened in the curved space-times, that will have a dierent meaning depending on the space-time we consider. Achronal ANEC for maximally symmetric space-times: Let us rst show how (2.19) be- comes the achronal ANEC for de Sitter and anti-de Sitter. Since the conformal factor in these cases is independent of (see table 2.2) we can remove the weight function from the integral (2.19). Given that both dS and AdS are maximally symmetric space-times, the vacuum stress tensor is proportional to their respective metrics (see appendix A.1), so that the additional contribution in (2.19) vanishes hT i 0 /g =) hT i 0 /g dv d dv d = 0 : (2.20) The ANEC is not sensitive to the non-zero vacuum energy of (A)dS since we must project along the null direction. Overall, the constraint in (2.19) becomes E (A)dS (~ x ? ) Z +1 1 d T 0 ; (2.21) that is the achronal ANEC for both dS and AdS. When making this identication, it is crucial that the null geodesics are complete and an ane parameter in (A)dS. Constraint for incomplete and achronal geodesics: The mapping of the Minkowski ANEC (2.19) gives a dierent type of constraint when applied to RS d1 and AdS 2 S d2 . To start, the conformal factor depends on so that it cannot be removed from the integral. Changing the integration variable to the ane parameter (dened in table 2.2 for each case) we obtain the same condition for both cases Z =2 =2 d cos d ( ) T h T i 0 0 ; (2.22) where we remember to consider the implicit factors ofd in the denition ofT when changing the integration variable. In this case, the vacuum stress tensor does not vanish and can be computed 18 from the results in [73]. For the case of the Lorentzian cylinder, it is given by RS d1 : h T i 0 = 8 > > < > > : 4(1) d=2 A d (d 1)Vol(S d ) ; d even ; 0 ; d odd ; (2.23) with A d the trace anomaly coecient (we are following the conventions in [74]). It is quite clear the constraint in (2.22) is not the achronal ANEC. Not only it is along an achronal but incomplete null geodesic, but there is also a non-trivial weight function cos d ( ). In the integration rangej j=2, the function cos d ( ) is non-negative, smooth and vanishes at the end points. The rapid decay of the function atj j = =2 is crucial, given that it is precisely at the boundary of a sharply integrated operator, where large amounts of negative energy can accumulate. 4 Since there are no complete and achronal null geodesics inRS d1 , (2.22) provides a novel condition in a setup where the achronal ANEC simply does not apply. 5 For AdS 2 S d2 the situation is slightly dierent, given that there are other complete and achronal null geodesics that move between the two AdS 2 boundaries. The conformal transformations studied in this section are not useful for arriving at the achronal ANEC in AdS 2 S d2 for these geodesics. 6 In the following chapter we shall directly prove the achronal ANEC for arbitrary QFTs in AdS 2 S d2 . 2.2 Monotonicity of relative entropy In this section we investigate whether some of the results in the previous section can be obtained using a dierent method, which uses the monotonicity property of relative entropy. The idea is to use the same approach as in [62], where the Minkowski ANEC was derived from relative entropy. Let us start with a brief review of the approach used in that paper. Consider a smooth curve in the null plane (2.4) dened by =A(~ x ? ) which splits the surface in two regionsN plane =A + [A , whereA are given by A(~ x ? ). Given a QFT in d- dimensional space-time X we take the space-time regionDA + for whichA + is its future horizon, and analogously forDA . A diagram of the setup is given in gure 2.4. For these space-time 4 See section 4.2.4 of [75] for an explicit example of this feature in two dimensional CFTs. 5 The condition (2.22) has been recently derived for holographic CFTs inRS d1 dual to Einstein gravity [76]. 6 While the geodesics (2.7) withj~ x ? j = 0; +1 reach the AdS2 boundaries, the mapping of the operator (??) breaks down due to an overall factor 1=j~ x ? j coming from w 2 (;~ x ? ). 19 Figure 2.4: Plot ofDA + (green) andDA (red) in the (T;X) plane in Minkowski space-time. The function A(~ x ? ) (blue dot) changes along the null plane XT = 0 as we move through the transverse directions~ x ? outside of the page. A very nice 3D picture of this region can be found in gure 1 of [1]. regions let us consider the reduced density operator A associated to the vacuum statej0i. We can dene A as the operator which satises the following property h0jO A j0i = Tr A O A ; (2.24) forO A any operator (not necessarily local) supported exclusively in DA . Given a reduced density operator its logarithm denes the modular Hamiltonian K A = ln( A ) + const, where the constant is xed by normalization. For this setup the modular Hamiltonian of the vacuum state was computed in [70] (see also [62, 67, 77]) and shown to have the following simple local expression K A =2 Z A dS (A(~ x ? ))T (;~ x ? ) ; (2.25) where dS = d~ x ? d is the induced surface element on the null plane and T is dened in (2.15). WhenA(~ x ? ) = 0 the regions in gure 6.3 corresponds to the Rindler wedge and its complement, so that (2.25) follows from the Bisognano-Wichmann theorem [78]. In this case the modular Hamil- tonian can be written as a local integral over any Cauchy surface inDA , not necesarily along the null horizons. This is not true when A(~ x ? ) is a non-trivial function, since the operator has a local expression only along the null surfaceA [70]. It is useful to also consider the full modular Hamiltonian ^ K A +, dened for a generic space-time region V as ^ K V =K V K V c ; (2.26) 20 where V c is the causal complement of V . Using (2.25) we nd ^ K A + = 2 Z N plane dS (A(~ x ? ))T (;~ x ? ) ; (2.27) where the integral is now over the full null plane. This operator has the advantage that it is globally dened in the Hilbert space, without any ambiguities that can arise in (2.25) from the boundary of integration. In the context of the Tomita-Takesaki theory ^ K V determines the modular operator. To prove the Minkowski ANEC, [62] combined the full modular Hamiltonian in (2.27) together with relative entropy, that is dened as S(jj) = Tr ( ln()) Tr ( ln()) 0 ; (2.28) where and are any two density operators. The monotonicity property of relative entropy implies that given any two space-time regions such that AB, the reduced operators satisfy the inequal- ity S ( A jj A ) S ( B jj B ). Taking as a pure state and starting from this inequality and an analogous one for the complementary regions, it is straightforward to prove following constraint [79] ^ K A ^ K B 0 ; for AB ; (2.29) where ^ K A=B is the full modular Hamiltonian of . 7 Using (2.27) we can explicitly write the inequality for null deformations of Rindler, which gives ^ K A ^ K B = 2 Z R d2 d~ x ? (B(~ x ? )A(~ x ? )) Z +1 1 dT (;~ x ? ) 0 : Since this condition must hold for arbitrary regions such that B(~ x ? )A(~ x ? ) 0, we conclude the operator between square brackets must be positive, which proves the ANEC for arbitrary QFTs in Minkowski [62]. Our strategy for extending this proof is simple. Using conformal transformations we map the modular Hamiltonian in (2.27) to (A)dS and the Lorentzian cylinder. From this we can explicitly write the inequality (2.29) coming from relative entropy and obtain a bound for the energy along 7 The inequality implied by relative entropy is more general than (6.1) and given by Tr ^ KA ^ KB 2S f (A;B) ; whereS f (A;B) is the free entropy of the state. This is a non-negative and UV nite quantity constructed from the entanglement entropy 2S f (A;B) = (SAS A 0) (SBS B 0), see [79]. If is a pure state, the free entropy vanishes and we recover (2.29). 21 null geodesics. We shall see that this procedure is non-trivial and while it works for de Sitter and the Lorentzian cylinder, it fails to give the ANEC in anti-de Sitter. Along the way we obtain several new modular Hamiltonians and compute their associated entanglement entropies. 2.2.1 Taking the null plane on a conformal journey - Take II Since our aim is to map the modular Hamiltonian (2.25), given by an integral over a region of the null plane, we start by discussing the geometric transformation of the null plane. Although we have already analyzed this in the previous section, the resulting surface (2.7) has a complicated coordinate description which is not the most convenient. We now consider a dierent conformal transformation that is more useful for writing the transformed modular Hamiltonian. Instead of mapping the null plane directly to the cylinder, we rst consider a conformal trans- formation mapping the Minkowski space-time X = (T;X; ~ Y ) into itself x = (t;x;~ y ). The transformation is given by x (X ) = X + (XX)C 1 + 2(XC) + (XX)(CC) D ; (2.30) where (XX) = X X . This gives a space-time translation in the D = (1=2; 1=2; ~ 0 ) direction together with a special conformal transformation with parameter C = (0; 1; ~ 0 ). The Minkowski metric in the new coordinates becomesds 2 =w 2 (x ) dx dx , where the conformal factor is given by w 2 (X ) = 1 + 2(XC) + (XX)(CC) 2 : Evaluating this along the null plane (2.4) we nd w 2 (;~ x ? ) = 4 ( +p(~ x ? )) 2 where p(~ x ? ) = 1 2 j~ x ? j 2 + 1 : (2.31) The mapped surface can be found by evaluating (2.30) in the parametrization of the null plane (2.4) x (;~ x ? ) = 1 2 p(~ x ? ) +p(~ x ? ) 1;~ n(~ x ? ) ; ~ n(~ x ? ) = j~ x ? j 2 1 j~ x ? j 2 + 1 ; 2~ x ? j~ x ? j 2 + 1 ; (2.32) where ~ n2R d1 is a unit vectorj~ n(~ x ? )j = 1. This surface corresponds to a future and past null cone starting from the origin x = 0. Although is not ane anymore, we can dene an ane 22 parameter according to () =p(~ x ? )(1=(2) 1), so that the surface is given by x (;~ x ? ) =(1;~ n(~ x ? )) ; (;~ x ? )2RR d2 : (2.33) Positive corresponds to the past null cone of the origin x = 0, while negative gives the future cone. The transverse coordinates ~ x ? parametrize a unit sphere S d2 in stereographic coordinates, as can be seen by computing the induced metric on the surface (2.33) and obtaining 2 d 2 d2 (~ x ? ). There is a subtlety in this transformation that we must be careful with. As we can see from the description in terms of in (2.32), there is a discontinuity in the mapping when =p(~ x ? ), that is precisely where the conformal factor (2.31) vanishes. Similarly to the previous mapping to AdS d in table 2.2, this is signaling a failure of the transformation, which is somewhat expected given that special conformal transformations are not globally dened in Minkowski but on its conformal compactication, the Lorentzian cylinder. To properly interpret the surface (2.33) we must go to the cylinder. Since a single copy of Minkowski is not enough to cover the whole cylinder, we consider an innite number of Minkowski space-timesM n andM m labeled by the integers (n;m), so that the whole cylinder manifoldM cylinder is obtained from M cylinder = [ n2Z M n [ [ m2Z M m : To each of the Minkowski copies we apply a slightly dierent conformal transformation r (n) ( ) = tan( =2) ; r (m) ( ) = tan( =2) ; (2.34) where the domain of the coordinates in each case is given by D n = 2R : ( 2n)2 [;] ; 2 [0;] ; D m = 2R : ( (2m 1))2 [0; 2] ; 2 [0;] : (2.35) In every case, the transformations act in the same way as in table 2.1 but mappingM (n;m) to dierent sections of the Lorentzian cylinder. These are given in the (;) plane by the shaded blue and orange regions in the second diagram from the left in gure 2.5. The main dierence between the n (blue) and m (orange) patches is that the n series maps the Minkowski origin to the North pole, while for m the origin is mapped to the South. 23 North Pole South Pole AdS center AdS boundary AdS boundary AdS boundary Figure 2.5: Mapping of the null plane in Minkowski to the null surfaces in the (;) plane. The dierent diagrams indicate the sections of the cylinder covered by each of the space-times. We see that discontinuity at =p(~ x ? ) in the Minkowski case is nothing more than the surface going from the Minkowski patchM n=0 toM m=0 in the second diagram. For de Sitter we consider a slight variation of the Weyl rescaling so that the surface ts in the space-time, see discussion above (2.40). Let us now use these relations to map the null plane across the special conformal transformation and into the cylinder. From (2.32) we can write the null radial coordinates r on the surface as r (;~ x ? ) = p(~ x ? ) j +p(~ x ? )j ( +p(~ x ? )) : (2.36) Applying the transformation associated to the patch n = 0 in (2.34) to the region of the null plane >p(~ x ? ) and m = 0 to p(~ x ? ) =) r (n=0) : + = 0; tan( =2) = p(~ x ? ) +p(~ x ? ) ; 2 [0;] ; 0 corresponds to the bulk interior. The d-dimensional metric g (z;v) admits an expansion in powers of z given by [83] g (z;v) =g (0) (v) +z 2 g (2) (v) + +z d ln(z 2 =L 2 )h (v) +z d g (d) (v) +o(z d ) ; where h is non-zero only for even d and o(z d ) means terms that vanish strictly faster than z d . The rst term in this expansion g (0) gives the space-time in which the boundary CFT is dened. Since in this case we are interested in a de Sitter background, we have from (2.3) g (0) dv dv = d + d + sin 2 ()d 2 (~ v ) sin 2 [( + )=2] ; (2.63) wherev = ( + ; ;~ v ) with the null coordinates =. We have written dS with the conformal factor sin 2 () so that 2 [; 0]. The higher order terms h and g (n) (v) with n < d can be obtained by perturbately solving Einstein's equations. They are all written in terms of geometric quantities built from the boundary metricg (dS) [83], i.e. they are a complicated functions of the Riemann, Ricci and curvature tensor of g (dS) and their covariant derivatives. For instance, the rst order term is given by g (2) = 1 d 2 R 2(d 1) g (dS) R ; (2.64) where the Ricci tensor and scalar on the right-hand side are computed from the metricg (dS) . Given that in this particular case we are considering a de Sitter boundary, we can use the fact that it is maximally symmetric, so that the Riemann tensor is completely xed by the metric R =g g g g : From this we see that (2.64) is proportional to the boundary metric g (2) =g (dS) =2. The powerful observation is that this is true for all the higher order terms h and g (n) with n<d. 11 Although the actual proportionality constants m n cannot be computed for arbitrary d, it will be enough to use that they are proportional h =m d g (dS) ; g (n) =m n g (dS) ; n<d : Using this, we can write any asymptotically AdS metric with a de Sitter boundary as ds 2 = (L=z) 2 h dz 2 + m(z)g (dS) (v) +z d g (d) (v) dv dv +o(z d ) i ; (2.65) 11 Since the Riemann is proportional to the metric, the terms in g (n) involving covariant derivatives vanish. 33 where the function m(z) satises m(z = 0) = 1 and is determined from the coecients m n m(z) = 1 +m 2 z 2 + +m d z d ln(z 2 =L 2 ) : This expansion to order o(z d ) will be enough for our purposes. The higher order terms are determined by the particular state in the boundary CFT. The rst undetermined contribution g (d) is related to the expectation value of the stress tensor of the dual CFT according to the standard AdS/CFT dictionary hT i = dL d1 16G g (d) (v) +X [g (n<d) ] ; (2.66) where X gives the anomalous term of the stress tensor in the CFT and G is Newton's con- stant. Although in a general setup X is a functional of g (n) with n < d, we can use the same observation as before to conclude that the anomalous terms is also proportional to the boundary metric X =x d g (dS) . If we project the stress tensor along the null direction , the anomalous terms drops out and we nd hT i = dL d1 16G g (d) (v) ; g (d) = dv d dv d g (d) : (2.67) 2.3.2 Curve ansatz and no bulk shortcut Let us now describe the setup that will allow us to obtain the ANEC. Consider a null geodesic in the boundary moving along the direction v ( ) = (0; ;~ v ) ; 2 [ 0 ; + 0 ] ; (2.68) where ~ v is xed and the null tangent vector is given by (0; 1; ~ 0 ). The parameter 0 2 [0;] determines the initial and nal points of the geodesic. For 0 = the geodesic is complete, going from the South pole of de Sitter at past innity to the North at future innity, while for 0 = 0 it is a single point. In the left diagram of gure 2.7 we sketch this curve in blue in the (;) plane. Although is not an ane parameter in dS, it is convenient to describe the geodesic in this way. We now wish to construct a bulk curve which starts at the same point as (2.68) at the boundary, goes into the bulk and ends in some other point at the boundary (not necessarily the same one as (2.68)). Consider the curve given by x A ( ) = (z;v ) = (f z ( );f + ( ); ;~ v ) ; 2 [ 0 ; + 0 ] ; 34 Figure 2.7: On the left we have a diagram of the setup in the (;) plane, with the shaded blue region corresponding to de Sitter. The boundary curve is shown in blue, while in green and red we show to dierent bulk curves going out of the page which satisfy the boundary conditions (2.70). The no bulk shortcut property implies that if these bulk curves are causal, the red trajectory is forbidden. On the right we plot our ansatz for the functionf z ( ) which shows how the bulk curve goes into the bulk for dierent values of L. As ! 0 the depth of the curve decreases and since 0 = d1 , its range in the boundary goes to 2 [0; 2]. which has a tangent vector equal to dx A d = (f 0 z ( );f 0 + ( ); 1; ~ 0 ) = (f 0 z ( );k ( )) : (2.69) The functions f z ( ) and f + ( ) must satisfy the following boundary conditions f z ( 0 ) = 0 ; f + ( + 0 ) = 0 ; (2.70) which ensures that the bulk curve behaves in the way we just described. A sketch of two bulk curves in red and green are shown in the left diagram of gure 2.7. The nal position of this curve in the boundary is determined from nal + =f + ( 0 ). The no bulk shortcut property is the statement that there is no bulk causal curve x A ( ) whose end point at the boundary is at the past of the end point of (2.68). More concretely, it implies the following condition If g AB dx A d dx B d 0 =) nal + =f + ( 0 ) 0 ; (2.71) whereg AB is the full bulk metric given in (2.65). This forbids a causal curve as the red one shown in the left diagram of gure 2.7. Violation of the no bulk shortcut property would result in causality and locality problems of the boundary theory (see [27, 71, 76, 84] for related discussions). 35 The strategy is to construct a particular causal bulk curve given in (2.69), such that the no bulk shortcut property (2.71) gives the ANEC for the boundary theory. From the expansion of the bulk metric in (2.65), the curve is causal as long as it satises the following constraint (f 0 z ( )) 2 + h m(f z ( ))g (dS) ( ) +f z ( ) d g (d) ( ) i k ( )k ( ) +o(z d ) 0 ; (2.72) wherek ( ) is given in (2.69). We will consider a particular bulk curve whose maximum depth in the bulk is given byL with a dimensionless quantity, and expand to leading order in 1. This curve must satisfy the boundary conditions in (2.70) to every order in as well as the causality constraint (2.72) to leading order. Our ansatz is inspired by the calculations in [71, 76]. For the function giving the z coordinates f z ( ) we choose f z ( ) =L tan( 0 =2)j cot( =2)j tan( 0 =2) : We plot this in the right diagram of gure 2.7 for several values of . The function is positive in the range of 2 ( 0 ; + 0 ) and vanishes at the end points, so that it satises the boundary conditions (2.70). The maximum depth of the curve into the bulk is given by L. To obtain the ANEC we relate the parameter 0 to according to 0 = d1 : (2.73) The limit of 1 then corresponds to a bulk curve near the boundary which covers a complete null geodesic in de Sitter (2.68), see Fig 2.7. If we expand for small we nd f z ( ) = j cot( =2)j 2 d +O( 2d ) : From this we see that the function f z ( ) is of order while its derivative goes like d . This is one of the crucial properties of the ansatz, since it ensures that the rst positive term in the causality constraint (2.72) is subleading in the expansion. For the remaining function f + ( ) we consider the following f + ( ) = d Z + 0 d 0 sin 2 ( 0 =2)g (d) (0; 0 ;~ v ) +Q( ) tan( 0 =2) + cot( =2) tan( 0 =2) d+ ; where Q( ) is any regular function and a small positive number. This function satises the boundary condition in (2.70) since it vanishes at the initial point = + 0 . From this, we can 36 write the tangent vector for the boundary components k ( ) in (2.69) in an expansion in as k ( ) = dv d + h sin 2 ( =2)g (d) (0; ;~ v ) d +Q 0 ( ) d+ i (1; 1; ~ 0 ) +O( d+1 ) : (2.74) Now that we have a bulk curve which satises the boundary conditions in (2.70), we check that it is causal to leading order in , i.e. that it satises (2.72). Expanding this constraint we nd m(f z ( )) sin 2 [(f + ( ) )=2] h sin 2 ( =2)g (d) (0; ;~ v ) d +Q 0 ( ) d+ i + d g (d) (0; ;~ v ) +o( d ) 0 ; where we have used that the de Sitter metric is given by (2.63) so that the contraction of the term du =d vanishes. Using that the function m(f z ( )) = 1 +O( 2 ) and expanding the sine in the denominator we nd g (d) (0; ;~ v ) d + Q 0 ( ) sin 2 ( =2) d+ + d g (d) (0; ;~ v ) +o( d ) 0 : The leading order in d involving the metricg (d) cancels and the rst non-vanishing contribution is given by d+ . Recall that o( d ) means terms that vanish strictly faster than d . This means that for any xed bulk space-time (corresponding to a state in the boundary CFT) we can x > 0 to be small enough so that it is the leading contribution in when compared to the unknown terms o( d ). In this way, the causality constraint reduces to the following condition on the functionQ( ) Q 0 ( ) 0 : (2.75) By xing this function such that it satises this property we are guaranteed to have a causal curve. Now that we have constructed the bulk causal curve, we can investigate the consequences of imposing the no bulk shortcut property in (2.71). Writing this explicitly we nd nal + = d Z + 0 0 d 0 sin 2 ( 0 =2)g (d) (0; 0 ;~ v ) + 2Q( 0 ) d+ 0 : Since the bulk curve is causal only in the limit of 1 we must expand in . Doing so, and using that the boundary stress tensor is related to g (d) according to (2.67) we nd Z 2 0 d 0 sin 2 ( 0 =2)hT (0; 0 ;~ v )i 2dL d1 16G lim !0 h Q( d1 ) +::: i : There are three possibilities for the value of the function Q( ) as ! 0. The least interesting case is when it diverges to +1 faster that goes to zero so that the bound becomes trivial. On 37 the contrary, if it diverges to1 then the causality condition Q 0 ( ) 0 in (??) is not veried and the curve is not causal. The most interesting case is when Q( ) goes to a constant value, so that the right hand side vanishes and we obtain a non-trivial condition given by Z 2 0 d 0 sin 2 ( 0 =2)hT (0; 0 ;~ v )i 0 : This is actually the ANEC in de Sitter, as can be seen by remembering that the parameter is not ane. If we change the integration variable to an ane parameter ( ) = cot( =2) (see table 2.3 noting that = =2), we obtain the achronal ANEC Z +1 1 dT (;~ v ) 0 : 38 Chapter 3 Achronal ANEC for general QFTs In the previous chapter we have shown how the achronal ANEC can be proven for arbitrary CFTs in de Sitter, anti-de Sitter and AdS 2 S d2 . The aim of this chapter is to show how these results can be extended to arbitrary QFTs that are not necessarily conformal. We do this is by computing the vacuum modular hamiltonian of a null deformed region to rst order in the deformation parameter and then using monotonicity of relative entropy. This approach was rst applied to arbitrary QFTs in Minkowski [62] and, as shown in section 2.2, also works for CFTs in certain curved backgrounds. The three sections of this chapter, sections 3.1, 3.2 and 3.3, deal with the AdS 2 S d2 , dS d and AdS d cases respectively. 3.1 Near extremal horizons In this section we prove the achronal ANEC for arbitrary QFTs in AdS 2 S d2 . Let us rst start by reviewing how this space-time arises from the near horizon geometry of near extremal black holes, which is the main reason we are actually interested in AdS 2 S d2 . For concreteness, consider four dimensional Einstein-Maxwell theory with vanishing cosmological constant I[g ;A ] = 1 16 Z d 4 x p gR 1 4 Z d 4 x p gF F : The extremal and spherically symmetric black hole solution is given by [85] ds 2 =f(r)dt 2 + dr 2 f(r) +r 2 d 2 2 ; where f(r) = rr h r 2 ; (3.1) 39 Figure 3.1: The left diagram corresponds to the Penrose diagram of the exterior region of the extremal black hole in the coordinates (=r h ;) (3.3). The Minkowski null innityI is indicated in green, while the future and past horizon of the black holeH in blue. Several xedr trajectories are plotted in gray. The right diagram gives the maximally extended space-time, with the time-like singularity in red. andr h > 0 is an integration constant which determines the horizon radius (related to the mass and charge of the black hole). The line element on the unit sphere S 2 is given by d 2 . To analyze the causal structure we rst dene the tortoise coordinate r (r) dr dr = 1 f(r) =) r r h = r h rr h + 2 ln r h rr h r r h ; (3.2) where r (r)2 R. The advantage of this coordinate is that the (t;r ) sector is conformally at. Further dening the coordinates (;) asr t =r h tan( =2) withj j =j=r h j, we nd ds 2 = f(r(r ( ))) 4 cos 2 ( + =2) cos 2 ( =2) d 2 +r 2 h d 2 +r(r ) 2 d 2 2 : (3.3) The Penrose diagram of the outer region of the extremal black hole is obtained by taking xed values in theS 2 and disregarding the conformal factor in (;). This give the diamond in the (;) plane seen in the left diagram of gure 3.1. The future and past black hole horizonsH are given by = , while the asymptotic null innity of Minkowski by =. In gray we plot some constant r curves. The maximal extension of the space-time is shown in the right diagram of this gure. The near horizon limit is obtained by taking (rr h )r h , where the tortoise coordinate r (r) 40 AdS Boundary AdS Boundary Figure 3.2: Penrose diagram of the near horizon limit of the exterior region of an extremal black hole, which corresponds to the Poincare patch of AdS 2 . The boundaries are located at = 0; while the black horizonH at =. We have plotted several constant r curves in gray. To the right we have the maximal extension, corresponding to global AdS 2 . in (3.2) is particularly simple r (r) =r 2 h =(rr h ) and the black hole metric in (3.3) becomes ds 2 ' d 2 +r 2 h d 2 sin 2 () +r 2 h d 2 2 AdS 2 S d2 ; (3.4) where we recognize an AdS 2 factor in global coordinates. Since in this limitr (r) only takes positive values, the coordinate is now restricted to2 (0;), with = 0; corresponding to the two AdS 2 boundaries. In the left diagram of gure 3.2 we plot the AdS 2 factor of the metric, with some constant r trajectories in gray. The near horizon geometry of the extremal black hole corresponds to the Poincare patch of AdS 2 , with the black hole horizonH given by the Poincare horizon. The near horizon geometry can be maximally extended to the right diagram of gure 3.2, which is global AdS 2 . Comparing with the full black hole Penrose diagram in gure 3.1, we see the near horizon limit corresponds to cutting the diagram in half, placing one AdS 2 boundary at the singularity and the other in the middle. 1 Since the study of QFTs in the full black hole background in (3.1) is complicated, it is useful to consider the simpler near horizon instead, hoping the most relevant quantum aspects of the horizon are captured in this approximation. An analogous calculation for the extremal black hole in arbitrary dimensions shows the near 1 In a slightly dierent way, the space-time AdS2S d2 also arises from the near horizon limit of near extremal black holes, see [2]. 41 horizon metric is given by ds 2 r 2 h = d 2 +d 2 sin 2 () +d d2 (~ v ) ; d 2 d2 (~ v ) = 4jd~ vj 2 (1 +j~ vj) 2 ; (3.5) where the unit sphere S d2 is parametrized in stereographic coordinates ~ v2R d2 . We have also conveniently rescaled the time coordinate ! =r h . Complete achronal null geodesics in this space-time correspond to paths going between the two AdS 2 boundaries with xed coordinates on S d2 . These geodesics can be described in terms of an ane parameter as v (;~ x ? ) = ( + ; ;~ v ) = (=2; 2 arccot()=2;~ x ? ) ; (;~ x ? )2RR d2 : (3.6) The null surface obtained from ~ x ? 2R d2 goes between the AdS 2 boundaries at ==2 and coincides withH + in gure 3.2 after a rigid time translation. 3.1.1 Non-conformal free scalar In this subsection we consider the achronal ANEC AdS 2 S d2 for the simplest non-conformal QFT: a massive scalar eld non-minimally coupled to the metric. We follow the simple approach used in [86] for Minkowski, which involves showing the ANEC operatorE(~ x ? ) can be written as E(~ x ? ) =WW y 0 for some operator W . An arbitrary free scalar in a curved manifold is characterized by the action I[g ;] = 1 2 Z d d x p g g (@ ) (@ ) + m 2 0 +R 2 ; (3.7) where m 0 is the bare mass and the non-minimal coupling to the space-time geometry. Varying the action with respect to the metric it is straightforward to compute the stress tensor and nd T = (@ ) (@ ) + (R r r ) 2 +g (2 1=2) h (@ ) (@ ) + (m 2 0 +R) 2 i ; (3.8) where we have used the equations of motion to write it in this way. The tangent vectork to the complete achronal null geodesics is obtained by dierentiating (3.6) with respect to . Projecting the stress tensor in the k direction, we can drop the third term. Since k only has non-trivial components in the AdS 2 sector, which is maximally symmetric, we haveR / g = 0. The covariant derivatives in the second term are replaced by ordinary 42 derivatives after using that k satises the geodesic equation k r k = 0. Putting everything together, the null component of the stress tensor is given by T (;~ x ? ) = 0 () 2 @ 2 () 2 : Integrating over 2R to get the ANEC operator, the second term gives a boundary contribution which drops and we nd E(~ x ? ) Z +1 1 dT (;~ x ? ) = Z +1 1 d 0 () 2 : (3.9) From this expression, we see that all classical free scalar elds in AdS 2 S d2 satisfy the achronal ANEC. This property is not obviously true after we quantize the theory, since the the operator 0 () 2 only makes sense after regularization, which spoils its positivity property. Given that the scalar is free and hermitian it can be written as (x ) = + (x ) + (x ) where + (x ) = (x ) y is expanded in terms of creation and annihilation operators as + (x ) = X i H i (x )a i ; h a i ;a y i 0 i = ii 0 : (3.10) The labeli goes over the linearly independent and orthogonal functionsH i (x ) solving the equations of motion. Using this expansion we can write the ANEC operator in (3.9) as E(~ x ? ) = Z +1 1 dhT i 0 + Z +1 1 d h (@ + ) y (@ + ) + (@ + )(@ + ) + h:c: i ; (3.11) where h:c: is the hermitian conjugate and we have identied the vacuum contribution as hT i 0 = h (@ + ); (@ + ) y i : This is divergent and requires a regularization procedure. After regularization, the vacuum con- tribution vanisheshT i 0 = 0 due to the general symmetry constraints explained in appendix A.1. While the rst term in the second integral (3.11) is an explicitly positive operator, the other is not. This means the ANEC holds if and only if the integral over of this additional term vanishes, which gives the following condition E(~ x ? ) 0 () C ii 0 = Z +1 1 dH 0 i ()H 0 i 0() = 0 ; 8 (i;i 0 ) : (3.12) 43 The proof of the ANEC is reduced to computing some integrals. Let us do this by rst writing H(x ) obtained from solving the equation of motion, given by r 2 AdS 2 +r 2 S d2 H(x ) = (m 2 0 +R)H(x ) : (3.13) Writing an ansatz withH(x ) =f(;)Y ` (~ v ) withY ` (~ v ), the eigenfunctions of theS d2 Laplacian, we nd r 2 AdS 2 f(;) = 2 f(;) ; where 2 = `(` +d 3) r 2 h + (m 2 0 +R) ; (3.14) with `2N 0 . The function f(;) satises the dierential equation of a scalar eld in AdS 2 with an eective mass 2 . The solution can be written as f n; (;) = sin () 2 F 1 n; +n; 1=2; cos 2 () e i(+2n) ; where n2N 0 and = 1 + p 1 + (2r h ) 2 2 1 : (3.15) To prove the achronal ANEC using (3.12) we can forget about the dependence ofH(x ) on theS d2 coordinates, since the complete achronal null geodesic in (3.6) has xed values of~ x ? . Moreover, it is convenient to translate the geodesics in (3.6) by redening !=2. Changing the integration variable in (3.12) to () = arccot() we get C (n;)(n 0 ; 0 ) = Z 0 d sin 2 ()f 0 n; (;)f 0 n 0 ; 0(;) ; (3.16) where the derivatives are now with respect to . We now to determine whether C (n;)(n 0 ; 0 ) = 0. To do so, let us start by considering the simple case in which n =n 0 = 0, so that the integral can be easily written and solved analytically as C (0;)(0; 0 ) = 0 Z 0 d sin() + 0 e i(2++ 0 ) = 0 : (3.17) For the rst few values of (n;n 0 ) the integral can still be solved analytically and shown to vanish, as we have explicitly checked for all combinations involving n;n 0 < 3. For higher values the analytic computation becomes very complicated and its convenient to integrate numerically. We have checked that the integral still vanishes with a numerical precision of up to eleven digits, for = 0 and all combinations (n;n 0 ) up to n;n 0 10 and for = 1=2 to = 20 in half-integer 44 steps. An analytic result which holds for arbitrary values of (n;n 0 ) can be obtained for the particular case of = 0 = 1, where the integral simplies to C (n;1)(n 0 ;1) = (1) n+n 0 (1 + 2n)(1 + 2n 0 ) Z 0 d sin 2 ()e i4(1+n+n 0 ) = 0 : We have also solved the integral numerically for random values of all the parameters and found always a vanishing answer. Overall, we have found enough analytic and numerical evidence to safely conclude the integral in (3.16) is identically zero, meaning the achronal ANEC holds for any free scalar in AdS 2 S d2 . While there might be a general analytic result showing the integral vanishes in full generality, we have not been able to nd it. Using the equations of motion to simplify and solve the integral has not been useful. An experimental observation is that after solving the indenite integral the result seems to always be given by sin() 1++ 0 g nn 0() where g nn 0() is a regular function at = 0;. The integral vanishes after evaluating at = 0;. 3.1.2 General QFTs Having shown the achronal ANEC holds for a non-conformal free scalar, in this subsection we attempt (and ultimately succeed) in proving the ANEC for arbitrary QFTs in AdS 2 S d2 . Our approach follows the derivation of the ANEC in Minkowski developed in [62]. The analogous region to the Rindler wedge in Minkowski for AdS 2 S d2 is given by D(A 0 ) = n (;;~ v )2R (0;)R d2 : + <=2 ; <=2 o ; (3.18) plotted in the left diagram of gure 3.3. More generally, we consider a deformation of this region in the null direction , parametrized by (~ v ) = (~ v ) as D(A) = n (;;~ v )2R (0;)R d2 : + <=2 ; <=2 + (~ v ) o ; (3.19) plotted on the right diagram of gure 3.3. For this setup, we compute the vacuum modular hamiltonian K A ln( A ) in a perturbative expansion in (~ v ). The technical result proven in the main part of this subsection is K A =K A 0 +r d2 h Z S d2 d (~ x ? ) (~ x ? ) Z +1 0 dT (;~ x ? ) +O( ) 2 ; (3.20) 45 Figure 3.3: On the left diagram we sketch the regionD(A 0 ) described in (3.18). The deformed regionD(A) shown in the right diagram contains the additional green section of size (~ v ). where T (;~ x ? ) is the stress tensor projected along the tangent vector dx =d to the geodesic in (3.6). The integration measure over the unit sphere S d2 is obtained from (3.5). From this expression the ANEC follows very easily. The rst order correction in the deforma- tion vector (~ v ) already contains \half" the ANEC operator. We can obtain the other half by constructing the full modular hamiltonian ^ K A K A K A c, which gives ^ K A = ^ K A 0 +r d2 h Z S d2 d (~ x ? ) (~ x ? )E(~ x ? ) +O( ) 2 ; (3.21) where we have dened the ANEC operatorE(~ x ? ) as E(~ x ? ) Z +1 1 dT : (3.22) Applying monotonicity of relative entropy (2.29) to this particular case gives the achronal ANEC ^ K A ^ K A 0 0 () D(A 0 )D(A) () (~ v ) 0 () E(~ x ? ) 0 : (3.23) In the remaining parts of this subsection we show that the modular hamiltonian is given by K A in (3.20) for arbitrary QFTs, completing the proof of the ANEC in AdS 2 S d2 . Undeformed region Let us start by computing the vacuum modular hamiltonian associated to the undeformed region D(A 0 ) in (3.18). We do this by following the same approach as the calculation for the Rindler 46 region in Minkowski, pedagogically explained in [87]. Since the computation requires a path integral formulation we rst analytically continue the metric (3.5) to Euclidean time E =i. This gives a useful representation of the vacuum state in terms of a path integral over the lower half E < 0 of the Euclidean space j0i = Z E <0 De I E [; E <0] = (3.24) where ( E ;x) represents all the elds in the theory. 2 The statej0i is a functional, which provides a number once we specify the boundary conditions of the path integral at E = 0 (green dashed line) according toh( E = 0 ;x)j0i. 3 Analogously, the hermitian conjugateh0j is given by the integral over the future region E > 0. Following standard arguments [88, 89], the reduced density operator corresponding to the vac- uum reduced to A 0 is obtained from the path integral by sewing the regions E < 0 and E > 0 but only for >=2. This gives the following path integral representation of A 0 A 0 = (3.25) Specifying the boundary condition at E = 0 + (red dotted line) gives the ket A 0 j( E = 0 + ;x)i, which after xing the eld conguration at E = 0 (green dotted line) gives a number h( E = 0 ;x 0 )j A 0 j( E = 0 + ;x)i. The path integral in (3.25) \propagates" the boundary con- ditions from either side of the cut along the Euclidean manifold, from the top red boundary to the bottom green one. Since the geometrical setup is very simple and there are no operator insertions in the path integral, we might hope that the \propagation" between the boundary conditions can be realized 2 The extra spatial directions parametrizing the S d2 are implicit in this pictorial description of the path integral. In describing the ground state as in (3.24) we are setting the vacuum energy to zero (by shifting the hamiltonian) and assuming no degeneracy. 3 There is no need to specify boundary conditions at = 0; since this is not a real boundary of the manifold, but a conformal boundary of Euclidean AdS2. Redening the radial coordinate according to% = cot() the regions = 0; are pushed to innity %!1. 47 Figure 3.4: On the left diagram we have the Euclidean ow generated by (3.26) applied on the surface at E = 0 + for 2 (0; 2). When we analytically continue to Lorentzian time E =i we obtain the ow on the right diagram. geometrically by an isometry of the space-time. This is certainly the case for the Rindler region in Minkowski, where such isometry simply corresponds to a rotation. Luckily, this is also the case in our setup, although the transformation is much less trivial and can be written as coth( E ()) = cosh( E ) cos() sinh( E ) + sin() cos() ; tan(()) = sin() cos() cos() sin() sinh( E ) ; (3.26) where 2 [0; 2] is the parameter in the transformation. The simplest way of obtaining this isometry is using the embedding description of Euclidean AdS 2 . It is also straightforward to check that the Euclidean metric (3.5) stays invariant. When applying this map to the surface E = 0 + in (3.25) we nd that for = 2 the surface is rotated in a non-trivial way to E = 0 (see left diagram in gure 3.4), doing precisely as we require by the path integral in (3.25). This means we can write the reduced density operator A 0 as A 0 =U( = 2) ; (3.27) whereU() is the unitary operator implementing the Euclidean transformation (3.26). Analytically continuing back to Lorentzian time =i E the isometry becomes cot((s)) = cos() cosh(s) sin() + sinh(s) cos() ; tan((s)) = sin() cosh(s) cos() + sinh(s) sin() ; (3.28) 48 where we have dened a real Lorentzian parameter si. The ow generated by this trans- formation for s2R is plotted in the right diagram of gure 3.4, where we see that its action is analogous to that of a boost in Minkowski: it is time-like in the wedge and mapsD(A 0 ) into itself. This is nothing more than the vacuum modular ow associated to the region. Using (3.27) we can write a concrete representation for A 0 as the operator implementing the transformation (3.28) with parameter s =i(2). The generator of the transformation is easily written in terms of the stress tensor T , so that we nd A 0 =U(s =2i) = 1 Z exp i (2i) Z dS T ; (3.29) whereZ is a normalization constant and the integral gives the conserved charge associated to (3.28). The integral is over a Cauchy surface inD(A 0 ) with future directed surface element dS . The Killing vector is obtained by expanding (3.28) for small s and has a simple expression when written in null coordinates = = cos( + )@ + cos( )@ : (3.30) Its magnitude vanishes at the boundaries ==2. Let us write the integral in (3.29) explicitly. Although we can choose any Cauchy surface in the regionD(A 0 ) (left diagram in gure 3.3) it is instructive to pick the future null horizon of D(A 0 ), which can be parametrized in terms of the ane parameter in (3.6) with 0. In this way we nd K A 0 = 2r d2 h Z S d2 d (~ x ? ) Z +1 0 dT (;~ x ? ) + const ; (3.31) where the constant comes fromZ in (3.29). In writing this expression we have chosen the canonical normalization for the future directed normal null vector, given byn =2 sin 2 () . The modular hamiltonian has a similar structure to that of the Rindler region in Minkowski. Before moving on to the null deformed region let us mention that it is straightforward to generalize this construction to a wedgeD(A 0 ) of any size, not necessarily =2, see appendix C of [5]. First order null deformation We now compute the rst order perturbation associated to the null deformation given in (3.19). We do so by adapting the methods used for Minkowski in [62, 90{92] to the curved background 49 AdS 2 S d2 . The basic idea is simple: since dealing with the vacuum reduced to the deformed regionD(A) is complicated, we apply a dieomorphism mappingD(A)!D(A 0 ). From (3.18) and (3.19) we see this is given by the transformation ~ = (~ v ), written covariantly as ~ x = x (~ v ). Since is not a Killing vector, the metric in the coordinates ~ x is no longer AdS 2 S d2 but is given by ~ g = @x @~ x @x @~ x g =g +L (g ) +O( ) 2 ; (3.32) where g is the AdS 2 S d2 metric andL (g ) = 2r ( ) the Lie derivative along . This approach allows us to trade the deformed regionD(A) in the space-time g by the simpler re- gionD(A 0 ) in the more complicated metric ~ g (3.32). The dieomorphism is implemented in the Hilbert space of the Euclidean theory by a unitary operator U(), that can be explicitly written as 4 U() = exp Z E =0 dS T (x) (~ v ) ; (3.33) where dS is the induced surface element on E = 0, with future directed unit normal. Splitting the integral over A 0 and A c 0 we can write it as U() = U A 0 U A c 0 . The reduced density operator A;g 5 in the deformed region is mapped by U A 0 according to A;g =U 1 A 0 A 0 ;~ g U A 0 = A 0 ;g + A 0 ;g ;U A 0 + A 0 ;g +O( ) 2 ; (3.34) where in the second equality we have expanded both U() and A 0 ;~ g to linear order in . While the rst two terms on the right-hand side are simple and can be written from (3.31) and (3.33), the third is more complicated. It can still be written explicitly using the path integral representation of A 0 ;g as (see equation (21) in [92]) A 0 ;g = A 0 ;g Z M E d d x p g (T hT i 0 )r ( ) ; (3.35) where the integral is over the Euclidean space-timeM E with a branch cut along2 (0;=2). The vacuum expectation value of the stress tensorhT i 0 comes from the variation of the normalization constant of A 0 ;~ g , and (to simplify notation) we shall leave implicit in what follows. 4 There is noi factor in the exponent since it is the Euclidean generator and no minus sign since the parameter in the dieomorphism is (~ v ). See [62, 91] for a path integral representation of U(). 5 The subscript g indicates the background metric. 50 An analogous perturbative expansion to (3.34) holds for the modular hamiltonian K A;g . This can be written explicitly using (3.35) and the smart identity proven in equation (2.14) of [90], 6 which results in K A;g =U 1 A 0 K A 0 ;~ g U A 0 =K A 0 ;g + K A 0 ;g ;U A 0 +K A 0 ;g +O( ) 2 ; (3.36) where K A 0 ;g = Z +1+i 1+i dz 4 sinh 2 (z=2) iz 2 A 0 Z M E d d x p gT (x)r ( ) iz 2 A 0 ; (3.37) with a free parameter in the range 2 (0; 2) that will be conveniently xed further ahead. We crucially get complex powers of the reduced density operator A 0 , which generates the modular ow of the undeformed regionD(A 0 ) (sketched in gure 3.4) on the Hilbert space. Using the conservation of the stress tensorr ( p gT ) = 0, the integral overM E can be reduced to its boundary @M E 7 K A 0 = Z +1+i 1+i dz 4 sinh 2 (z=2) Z @M E dS iz 2 A 0 T (x) iz 2 A 0 ; (3.38) where we drop the sub-index g for notation convenience. This is the non-trivial integral we must solve to compute the rst order contribution to the modular hamiltonian. The boundary @M E does not get contributions from the conformal boundary of Euclidean AdS 2 , but from the branch cut along E = 0 and 2 (0;=2) (red wiggly line in gure 3.5). 8 To describe the region @M E let us introduce two useful set of coordinates. First, we consider E =i E which upon analytic continuation yield the ordinary null coordinates =. In addition, it is also convenient to take the parameter 2 [0; 2] describing the Euclidean modular ow in (3.26) as a coordinate. An associated spatial coordinate which covers the entire Euclidean section can be chosen by parametrizing the surface ( E = 0;<=2). This corresponds to the red or green lines in gure 3.4, which propagate over the whole space-time as we vary 2 [0; 2]. A convenient parametrization of the initial surface at E = 0 is given by () = ln(cot(=2))2R 0 . 6 The Baker-Campbell-Hausdor formula used to derive equation (2.14) in [90] can be found in (4.173) of [93]. One must be careful with the sign convention of the Bernoulli numbers Bn, since they disagree between these references by a minus sign inB1. In the convention of [90] theBn are written in terms of the Riemann zeta function as Bn =n(1n), which can be written as an integral using (35) of [94]. 7 In appendix A.1 we show thatrhTi0 = 0, as required to obtain this relation. 8 What we usually call the AdS boundary is really a conformal boundary, meaning that the distance between any xed point and = 0; is always innity. When considering the region@ME we can drop contributions from = 0; in the same way we do for spatial innity in Minkowski. 51 Figure 3.5: Sketch of the boundary@M E obtained from the regionsR (in blue) andC (in green) described in (3.41) as;b! 0. The red wiggly line corresponds to the branch cut in the Euclidean space-time located at E = 0 and 2 (0;=2). From (3.26) we see that the coordinates (;) and ( E ;) are related according to 8 > < > : tanh( E ) = sin() tanh() cot() = cos() sinh() =) tan( E ) = cos()i sin() cosh() sinh() ; (3.39) so that the Euclidean metric is ds 2 E r 2 h = d E + d E sin 2 () +d 2 d2 =d 2 + sinh 2 ()d 2 +d 2 d2 : (3.40) Using these coordinates we can describe the boundary @M E , as the union of the surfaces C and R in the limit of ;b! 0 C = n (;)2 [0; 2]R 0 : 2 ; =b o ; R + = n (;)2 [0; 2]R 0 : = ; b o ; R = n (;)2 [0; 2]R 0 : = 2 ; b o ; (3.41) sketched in gure 3.5. We now proceed to compute the contributions of these surfaces to the integral in K A 0 (3.38). We refer to C and R as \branch point" and \branch cut" contributions respectively. Branch point contribution: Let us start by computing the branch point contribution, given by the integral over the (green) surface C in gure 3.5. The induced metric and inward pointing unit normal vector n are ds 2 r 2 h =b = sinh 2 (b)d 2 +d 2 d2 ; n = 1 r h = 1 r h @ E + @ + + @ E @ ; 52 where the derivatives in the normal vector can be easily computed from (3.39) @ E @ = 1 cos() cosh()i sin() : (3.42) Before using this to write the integral in (3.38), consider the action of the modular ow on the stress tensor, given by iz 2 A 0 T ( E ) iz 2 A 0 = @x @x @x @x T ( E (z)) ; (3.43) where the indices (;) correspond to E , while ( ; ) to the modular translated coordinates E (z). The complex parameter z along the integral in (3.38) can be written as z = s +i with s2R. In the (;) coordinates the modular ow is particularly simple and given by a rigid translation (z) = +iz, while is unchanged. Moreover, since 2 (0; 2) is a free parameter we can x it to = , so that the translated coordinate is (z) = is, purely imaginary and independent of . This crucially means the operator on the right-hand side of (3.43) is inserted only in Lorentzian time. For this value of the modular ow in the E coordinates can be found using (3.39) tan( E (z)) = cosh(s) sinh(s) cosh() sinh() ; (3.44) which is equivalent to the Lorentzian modular ow (3.28). The derivatives appearing in the Jacobian expression in (3.43) can be computed from @ E (z) @ E = @ @ E @ E (z) @ + @ @ E @ E (z) @ = cos() cosh()i sin() cosh(s) cosh() sinh(s) ; (3.45) where we used that E (z) in (3.44) is independent of . Putting everything together in (3.38) we write the contribution coming from the surface C as K A 0 C = 2r d2 h Z S d2 d (~ v ) (~ v ) Z +1 1 dsI(b;s) sinh(b)T ( E (z)) (cosh(b) cosh(s) sinh(s)) 2 + sinh(b)T + ( E (z)) cosh 2 (b) cosh 2 (s) sinh 2 (s) ; (3.46) where we have dened I(b;s) = Z 2 d 2 cosh(b) cos() +i sin() 4 sinh 2 ( s+i 2 ) = cosh 2 (b=2)e s (s)+sinh 2 (b=2)e s (s)cosh(b)(s): This integral is solved by taking the limit ! 0, changing the integration coordinate to w = e i and computing a residue. Using this in (3.46), the integral in s involving the component T of 53 Figure 3.6: Plot of the two kernels in the integral in (3.47) for several values of b. The left (right) diagram corresponds to the integral with positive (negative) s. the stress tensor is Z +1 0 ds sinh(b) cosh 2 (b=2)e s (cosh(b) cosh(s) sinh(s)) 2 T ( E (z)) + Z 0 1 ds sinh(b) sinh 2 (b=2)e s (cosh(b) cosh(s) sinh(s)) 2 T ( E (z)) ; (3.47) where we have omitted the contribution coming from the Dirac delta since it vanishes in the limit b! 0. Although the remaining terms also seem to vanish as b! 0, we must be careful since the integration region goes to innity. Plotting the two kernels in each integral for several values of b we obtain the plots in gure 3.6. The left (right) diagram corresponds to the integral with positive (negative) s. While the diagram in the right goes to zero as b! 0 for every value of s, the plot in the left contains a contribution that is not suppressed in the limit but merely translated to larger values of s. The position of the maximum is given by s max (b) = ln(coth(b=2)= p 3). Both integrals in s coming from the contributions to T + in (3.46) show the same behavior as the right diagram in gure 3.6, and therefore vanish in the limit b! 0. Thus, the only non-vanishing contribution to (3.46) is given by the rst integral in (3.47). To extract the surviving piece as b! 0, we redene the integration variable to u(s) = E (z) in (3.44) u(s) = arctan cosh(s) sinh(s) cosh(b) sinh(b) ; with inverse s(u) = ln 1 sin(u) tanh(b=2) cos(u) : Under this change of variables the rst integral in (3.47) becomes Z +1 0 ds sinh(b) cosh 2 (b=2)e s (cosh(b) cosh(s) sinh(s)) 2 T ( E (z)) = 1 2 Z u 0 (b) =2 du (1 + sin(u))T ( E =u; E + (u;b)) : whereu 0 (b) = arcccot(sinh(b)). The kernel is now independent ofb, meaning we can safely takeb! 0, which gives u 0 ! =2. Moreover, the complicated function E + (u;b) simplies to E + ! =2. 54 Putting everything together in (3.46) we nd that the only surviving contribution as ;b! 0 is K A 0 C =r d2 h Z S d2 d (~ v ) (~ v ) Z =2 =2 du (1 + sin(u))T ( E =u; E + ==2) : (3.48) Theu integral is over the future horizon of the undeformed regionD(A 0 ). Changing the integration variable to the ane parameter in (3.6) we nd K A 0 C =r d2 h Z S d2 d (~ x ? ) (~ x ? ) Z +1 0 dT (;~ x ? ) ; (3.49) where we recognize \half" the ANEC operator. Branch cut contribution: Let us now compute the two additional contributions coming from the integrals over the surfaces R in gure 3.5. The induced metric and unit normal vectors are easily found from (3.40) and (3.41) as ds 2 r 2 h = 0 =d 2 +d 2 d2 ; n = r h sinh() ; (3.50) where n corresponds to the (inward) unit normal vector to R . Similarly to (3.43) we can write the stress tensor insertions evolved under the modular ow, appropriately choosing the parameter so that the resulting operator is inserted at = 0. In each case we obtain the following expressions R + : i(s+i + ) 2 A 0 T ( =;) i(s+i + ) 2 A 0 = @x @x @x @x is 2 A 0 T ( = 0;) is 2 A 0 ; R : i(s+i ) 2 A 0 T ( = 2;) i(s+i ) 2 A 0 = @x @x @x @x is 2 A 0 T ( = 0;) is 2 A 0 ; (3.51) where + = and = 2. We have only applied the modular ow explicitly in the Euclidean direction, while left the ow in s still written in terms of a complex power of A 0 . The Jacobian matrices on the right-hand side of (3.51) are computed from (3.45) with s = 0 and = . Their behavior for small is always given by @ E (z)=@ E = 1 +O(), where higher orders in drop out since they do not contribute in the limit ! 0. This means the Jacobian matrices appearing in (3.51) are eectively equal to the identity and can be ignored. Putting everything together, the contributions from R to K A 0 in (3.38) are given by K A 0 R + =r d1 h Z S d2 d (~ v ) Z +1 b dn Z +1 1 ds 4 sinh 2 ( s+i 2 ) is 2 A 0 T ( = 0;) is 2 A 0 ; K A 0 R =r d1 h Z S d2 d (~ v ) Z +1 b dn Z +1 1 ds 4 sinh 2 ( si 2 ) is 2 A 0 T ( = 0;) is 2 A 0 ; (3.52) 55 where the overall sign dierence comes fromn + =n . While there is no easy way of solving each integral separately, the contribution of both regions R + [R results in a path in the complex s plane which simply picks the residue of the double pole at s = 0. This can be easily computed and written as K A 0 R + [R =r d1 h Z S d2 d (~ v ) Z +1 0 dn T ( = 0;);K A 0 : where we have safely taken the ;b! 0 limits. The unit normal vector n now points vertically upward in the Euclidean time direction @ E . The resulting integral only involves the stress tensor over the undeformed regionA 0 and we recognize it as the exponent of the unitary operator U A 0 () in (3.33), i.e. K A 0 R + [R = Z A 0 dS T (x) (~ v );K A 0 = U A 0 ;K A 0 : (3.53) All rst order contributions: We can now put everything together in (3.36) to obtain an explicit expression for the modular hamiltonian K A . Using (3.49) and (3.53) we nd K A =K A 0 +r d2 h Z S d2 d (~ x ? ) (~ x ? ) Z +1 0 dT (;~ x ? ) +O( ) 2 ; which is precisely the result given in (3.20) used to prove the ANEC. Notice that the terms coming from the branch cut R + [R in (3.53) crucially canceled the contributions from the variation of the unitary U A 0 in (3.36). This concludes the proof of the achronal ANEC for arbitrary QFTs in AdS 2 S d2 . 3.2 De Sitter In this section, we show how the same type of calculation allows us to prove the achronal ANEC for arbitrary QFTs in de Sitter. Taking global coordinates, the de Sitter metric is given by ds 2 L 2 = d 2 +d 2 + sin 2 ()d 2 d2 cos 2 () ; (3.54) whereL is the de Sitter length scale and2 [0;]. The time coordinate is constrained tojj<=2, with the spatial dS boundaries located at ==2, see the diagram in gure 2.1. Following the calculations in the previous section, we consider the vacuum statej0i and reduce it to the half-space A 0 at = 0. Since the topology of dS is that ofRS d1 , the half-space in this case corresponds 56 to the spherical cap given by 2 [0;=2]. Its causal domainD(A 0 ) is easily described in terms of the null coordinates = as D(A 0 ) = n (;;~ v )2 (=2;=2) [0;]R d2 : + <=2 ; <=2 o : This region is equivalent to the static patch of de Sitter. Its null deformation in the direction can be parametrized by the vector = (~ v ) as D(A) = n (;;~ v )2 (=2;=2) [0;]R d2 : + <=2 ; <=2 + (~ v ) o : (3.55) The diagrams of these regions are the same as the ones given in gure 3.3 for AdS 2 S d2 . The dierence is that the time coordinate is restricted tojj < =2 and there are no boundaries at = 0;, since in (3.54) these correspond to the North and South pole of the spatial S d1 . 9 Let us now compute the rst order contribution to the modular hamiltonian associated toD(A). Undeformed region: We start by considering the vacuum modular hamiltonian associated to the undeformed regionD(A 0 ). Analytically continuing to Euclidean time E = i the de Sitter metric (3.54) becomes ds 2 E L 2 = d 2 E +d 2 + sin 2 ()d 2 d2 cosh 2 ( E ) : (3.56) Since the function in the denominator does not vanish, the Euclidean time is free to take any real value E 2R. As in (3.24), the path integral over the region E < 0 gives a representation of the vacuum statej0i, so that the reduced density operator A 0 is obtained from the path integral with the boundary conditions given in (3.25). We should now look for an isometry of the Euclidean mani- fold (3.56) that smoothly maps between the surfaces 2 [0;=2) at E = 0 . Quite surprisingly, the appropriate isometry is exactly the same as in AdS 2 S d2 (given by (3.26)) whose action in the ( E ;) plane is shown in gure 3.4. It is straightforward to check that (3.56) is invariant under this transformation. The fact that the modular ow of both these space-times is the same is not trivial to us. 9 Since the time coordinate in dS is constrained tojj < =2 the regionD(A) for (~ v ) 0 intersects with the spatial boundary at ==2, see gure 3.3. The full modular hamiltonian we use to prove the ANEC is not aected by this. 57 The modular hamiltonian is obtained from the generator of the Lorentzian isometry (3.28) according to (3.29) K A 0 = 2 Z dS T + const ; = cos( + )@ + cos( )@ ; (3.57) where is a Cauchy surface onD(A 0 ). We can write this explicitly by taking as the future null horizon ofD(A 0 ), which can be parametrized as x (;~ x ? ) = ( + ; ;~ v ) = (=2; 2 arccot()=2;~ x ? ) ; (;~ x ? )2RR d2 ; (3.58) where for xed~ x ? the parameter is ane in the dS metric (3.54). This description coincides with the null surface in (3.6) for AdS 2 S d2 , where is also ane. While the coordinate description of the null curves coincides, the geodesics travel along very dierent space-times. From (3.57), the modular hamiltonian forD(A 0 ) is written along this null surface as K A 0 = 2L d2 Z S d2 d (~ x ? ) Z +1 0 dT (;~ x ? ) + const ; which has the same structure as for AdS 2 S d2 in (3.31). First order deformation: Given that the modular ow and ane parameter on de Sitter and AdS 2 S d2 coincide for the undeformed region, the computation of the rst order null deformation is almost exactly equivalent the one presented in the previous section. The series expansion in (~ v ) is given by (3.36) with K A 0 ;g in (3.38), which we rewrite here for convenience K A 0 = Z +1+i 1+i dz 4 sinh 2 (z=2) Z @M E dS iz 2 A 0 T (x) iz 2 A 0 : (3.59) Since the undeformed modular ow is the same, so is its action on the stress tensor T (x). The dierence comes from the integral @M E , which can be described in terms of the surfaces C[R (3.41) plotted in gure 3.5. Writing the Euclidean dS metric in the coordinates (;) in (3.39) we nd ds 2 E L 2 = sinh 2 ()d 2 +d 2 +d 2 d2 cosh 2 () : This diers only by the overall factor cosh 2 () with respect to the metric in AdS 2 S d2 (3.40), which contributes to dS in (3.59). 58 The integral over C in (3.41) is given by =b with b! 0. Since cosh(b)! 1, the calculation leading to (3.49) is identical and gives the same result K A 0 C =L d2 Z S d2 d (~ x ? ) (~ x ? ) Z +1 0 dT (;~ x ? ) : For the integral over the surfaceR (given essentially by = 0 ) the factor cosh 2 () contributes in a non-trivial way to give the induced surface element of dS at E = 0, which is dierent from that on AdS 2 S d2 . However, this is exactly what we require so thatK A 0 R + [R in (3.53) givesU A 0 , but with the unitaryU A 0 dened as in (3.33) on the surface E = 0 in de Sitter. This contribution is essential to cancel the commutator in (3.36), so that the end result for the null deformed modular hamiltonian in dS gives K A =K A 0 +L d2 Z S d2 d (~ x ? ) (~ x ? ) Z +1 0 dT (;~ x ? ) +O( ) 2 : (3.60) Using relative entropy, the same calculation leading to (3.23) gives the ANEC for a QFT in de Sitter. This modular hamiltonian agrees with the result for arbitrary CFTs to all orders in the deformation parameter in (2.55). 3.3 Anti-de Sitter An analogous construction can be considered for a QFT in anti-de Sitter. There is a more subtle aspect when it comes to nding an appropriate set of coordinates to describe the half-space region of AdS, given by half a cross section of the solid cylinder. The easiest way of parametrizing this region, is by considering some sort of spatial Cartesian coordinate~ x2R d1 , such that the boundary is located atj~ xj = 1. These coordinates can be dened from the embedding description of AdS, given by the surface (X 0 ) 2 (X 1 ) 2 + d X i=2 (X i ) 2 =L 2 ; (3.61) in the spaceR 2 R d1 ds 2 =(dX 0 ) 2 (dX 1 ) 2 + d X i=2 (dX i ) 2 : The constraint (3.61) is automatically satised if we dene the coordinates x = (;~ x ) as X 0 =L 1 +j~ xj 2 1j~ xj 2 sin() ; X 1 =L 1 +j~ xj 2 1j~ xj 2 cos() ; X i =L 2x i 1j~ xj 2 : (3.62) 59 where 2R and ~ x2R d1 withj~ xj < 1. The induced metric on R 2 R d1 gives the metric in global AdS ds 2 L 2 = 1 +j~ xj 2 1j~ xj 2 2 d 2 + 4jd~ xj 2 (1j~ xj 2 ) 2 : (3.63) The boundary is located atj~ xj = 1, with the interior of the solid cylinder described by the Cartesian coordinates ~ x. This unusual way of writing the AdS metric allows for a simple description of the half space. Picking an arbitrary direction in ~ x we write ~ x = (x;~ y ) with ~ y2 R d2 , so that the half-space A 0 at = 0 is given by A 0 = n (;x;~ y )2RRR d2 : x 2 +j~ yj 2 < 1 ; = 0 ; x> 0 o : A constant time surface for AdS 3 is plotted in gure 3.7, whereA 0 corresponds to thex> 0 region. Although the half-spaceA 0 has a very simple description in these coordinates, they are not well suited to describe its causal domainD(A 0 ), given that null geodesics in (3.63) are not given by simple straight lines. We can x this by applying another change of coordinates (x;j~ yj)! (; ), dened as 10 x = cos() 1 + sin() sin( ) ; j~ yj = sin() cos( ) 1 + sin() sin( ) ; with inverse cos() = 2x 1 + (x 2 +j~ yj 2 ) ; cot( ) = 2j~ yj 1 (x 2 +j~ yj 2 ) ; (3.64) where 2 (0;) and 2 (0;=2]. Taking stereographic coordinates ~ v2 R d3 to describe the angular direction of ~ y, the AdS metric (3.63) becomes ds 2 L 2 = 1 sin 2 ( ) d 2 +d 2 sin 2 () +d 2 + cos 2 ( )d 2 d3 (~ v ) : (3.65) Although hardly recognizable, this metric describes global AdS. There are two ways we can approach the boundary, obtained by taking ! 0 or! 0;, which from (3.64) corresponds tox 2 +j~ yj 2 ! 1 andx!1 respectively. Thex direction is mainly controlled by, with the half-spacex> 0 given by <=2. In gure 3.7 we plot the constant (; ) trajectories on a xed time slice of AdS 3 . The crucial aspect of (3.65) is that null curves in the (;) direction are straight lines. This is precisely what is needed to describe the causal domain of the half-space A 0 , that is given by D(A 0 ) = n (;; ;~ v )2R (0;) (0;=2]R d3 : + <=2 ; <=2 o ; 10 These relations are inspired by the holographic calculations of entanglement entropy in appendix B of [4]. 60 AdS boundary Figure 3.7: Constant time section of AdS 3 given by the region x 2 +y 2 < 1. We have plotted the constant and trajectories obtained from (3.64). The half space A 0 dened from x > 0 corresponds to <=2. where = . The transverse space is parametrized by ( ;~ v ), where ~ v 2 R d3 has one dimension less than in the previous cases in AdS 2 S d2 and de Sitter. It is now easy to consider the null deformed half-space, parametrized by = ( ;~ v ) as D(A) = n (;; ;~ v )2R (0;) (0;=2]R d3 : + <=2 ; <=2 + ( ;~ v ) o : (3.66) Plotting these regions in the (;) plane we obtain exactly the same diagrams as in gure 3.3, with the AdS boundary located at = 0;. Since the sector (;) of the AdS metric in (3.65) is exactly the same as for AdS 2 , the calculation of the vacuum modular hamiltonian associated toD(A) is extremely similar to that in AdS 2 S d2 . Let us sketch the calculation, highlighting the most salient dierences. Undeformed region: The vacuum modular hamiltonian is obtained in exactly the same way as for AdS 2 S d2 : we analytically continue to Euclidean time E = i, describe A 0 from the path integral in (3.25) and look for an isometry of the Euclidean manifold which generates the appropriate ow. Since the (;) dependence of the AdS metric (3.65) is the same as that of AdS 2 S d2 in (3.5), the appropriate isometry is the same as the one given by (3.26), which generates the ow shown in gure 3.4. The modular hamiltonian is related to the generator of this isometry (3.29) according to K A 0 = 2 Z dS T + const: ; = cos( + )@ + cos( )@ : (3.67) 61 We take as the future null horizon ofD(A 0 ), which we can parametrize in terms of an ane parameter 2R as x (; ;~ x ? ) = ( + ; ; ;~ v ) = (=2; 2 arccot()=2; ;~ x ? ) ; (3.68) where the transverse space is parametrized by ( ;~ x ? )2 (0;=2]R d3 . Notice that ~ x ? has one less component in comparison with the previous cases. The modular hamiltonian (3.67) along this surface can be written and we nd K A 0 = 2L d2 Z S d3 d (~ x ? ) Z =2 0 d cos( ) d3 sin( ) d4 Z +1 0 dT (; ;~ x ? ) + const : While the integral over the transverse space ( ;~ x ? ) is dierent, the sector is equivalent to the previous cases. First order deformation: The rst order contribution in the deformation parameter ( ;~ v ) in (3.66) follows exactly the same as for AdS 2 S d2 . The integral we must solve is again given by (3.59). The boundary of the Euclidean AdS manifold @M E written in the coordinates (;) in (3.39) is given by ds 2 E L 2 = d 2 + sinh 2 ()d 2 sin 2 ( ) + d 2 + cos 2 ( )d 2 d3 (~ v ) sin 2 ( ) : Since the (;) dependence of the metric is the same as for AdS 2 S d2 we can easily solve the integral in (3.59) and nd the following result for the null deformed modular hamiltonian on AdS K A =K A 0 +L d1 Z S d3 d (~ x ? ) Z =2 0 d cos( ) d3 sin( ) d4 ( ;~ x ? ) Z +1 0 dT (; ;~ x ? ) +O( ) 2 : (3.69) The crucial aspect of this relation is that we recover half the ANEC operator in . Writing the full modular hamiltonian ^ K A =K A K A c and using relative entropy, the same calculation leading to (3.23) gives the ANEC for a QFT in AdS for the null geodesics in (3.68). 62 Chapter 4 An application: Quantum BMS transformations So far we have shown the achronal ANEC holds for arbitrary QFTs in several curved manifolds. The aim of this chapter is to explore the consequences of the ANEC when studying quantum aspects of BMS transformations. Back in the sixties, Bondi, van der Burg, Metzner and Sachs (BMS) studied the symmetry algebra of asymptotically at space-times at future and past null innityI [95{97], and to their surprise, found that instead of the nite dimensional Poincar e algebra, space-time translations were enhanced to an innite dimensional sub-algebra they called supertranslations. It was later realized that by relaxing certain technical conditions, the Lorentz transformations could also be enhanced into what is commonly referred as superrotations [98{101]. Here, we call BMS algebra to the enhanced version containing both supertranslations and superrotations. 1 Innite- dimensional symmetry algebras should be taken seriously, as they have shown to play an important role in other setups, such as in the AdS 3 /CFT 2 [102] and Kerr/CFT [103] correspondence. In recent years, BMS asymptotic symmetries have been investigated from several dierent promising perspectives. Studies of the gravitational scattering matrix in Minkowski [104] have lead to interesting relations between BMS symmetry, soft theorems [105], and the so-called gravi- tational memory eects [106] (see [107] for a review and further references). The structure of BMS 1 In the four dimensional case there are two dierent innite dimensional extensions of the Lorentz algebra at null innity, one involving two copies of the Virasoro algebra and the other Di(S 2 ). In this work we shall mostly consider the extension involving Di(S 2 ). While a more appropriate term for these elements would be `super-Lorentz', we shall follow most of the literature and call them superrotations. 63 symmetry has also appeared at the horizon of classical black holes solutions [108, 109], and it has been suggested it supplies the necessary additional structure to provide a possible resolution of the black hole information paradox [110, 111]. In this chapter we carefully study quantum aspects of BMS symmetry in conformally at space- times, aiming towards possible applications to black hole physics and holography. To do so, we consider a simple system obtained by placing an arbitrary conformal eld theory (CFT) on a xed d-dimensional space-time. Using conformal symmetry, we are able to study BMS transfor- mations that are dened not only on the asymptotic null regions of MinkowskiI , but on certain Killing horizons H on a variety of conformally at space-times, including (A)dS, RS d1 and AdS 2 S d2 . 2 All the results of this chapter hold for arbitrary CFTs on any of these space-times and involve the following quantum aspects of BMS symmetry: the algebra satised by the charge operators, its representations on the Hilbert space of the CFT as well as the group action obtained by exponentiating the charges. For holographic CFTs that are well described by semi-classical Einstein gravity, we propose and provide evidence in favor of a holographic description of our CFT computations. Towards the end, we apply our construction to study some aspects of asymptotically at (near-)extremal black holes. Our analysis does not only involve the ordinary supertranslation and superrotation BMS trans- formation, but also a novel transformation that we call `superdilation'. We show how this asymp- totic transformation naturally arises when considering conformal theories and include it in all our analysis throughout this chapter. 3 Chapter summary We start in section 4.1 by describing a set of conformal transformations that map Minkowski to (A)dS d ,RS d1 and AdS 2 S d2 (see gure 4.1 for a sketch of some of their Penrose diagrams). The future null boundary of MinkowskiI + is mapped to a Killing horizonH + in the curved space- time, e.g. for AdS 2 S d2 the surface H + corresponds to the future boundary of the Poincar e patch of AdS 2 (left diagram in gure 4.1). All the results of this chapter apply to any of these 2 The BMS transformations we construct on the Killing horizonsH are closely related to previous work on horizon symmetries [108, 109]. 3 See [112{114] for previous studies on the relation between asymptotic and conformal symmetries. 64 space-times and their associated surfacesI + or H + . We then construct asymptotic Killing and conformal Killing vectors on the surface I + for Minkowski, and H + in the curved space-times. These vectors are a natural generalizations of translations, Lorentz transformations, special conformal transformations and dilations T (f); R (Y ) [ S (h); D (g) : (4.1) The rst two vectors depend on a function f and a vector Y A , and generate the ordinary BMS transformations, i.e. supertranslations and superrotations. The remaining vectors generalize special conformal transformations and dilations respectively and depend on two functions h and g. These are given in (4.17) for d = 3 and (4.24) for arbitrary dimensions, with the metric fall of conditions preserved by them nearI + (H + ) shown in tables 4.3 and 4.4. Evaluating the vectors (4.1) on the surfaceI + (H + ) we compute their algebra and nd that a subset of these vectors given by Asymptotic symmetries = T (f); R (Y ) [ D (g) ; (4.2) satisfy the closed algebra in (4.23) and (4.29). This is an extension of the ordinary BMS algebra that includes the superdilation vector D (g). It is the asymptotic transformations generated by the three vectors in (4.2) the one we study in the rest of the chapter. At this point we should issue a word of warning regarding the superdilation vector D (g), since the metric fall-o conditions preserved by this vector are more singular than those obtained for the ordinary BMS vectors T (f) and R (Y ). This means that we should be careful when studying superdilation transformations, as complications can arise when computing physical quantities as- sociated to D (g). More precisely, in the holographic analysis of section 4.4 we nd that certain conserved charges associated to D (g) diverge. Despite these issues we study superdilations throughout this chapter, since it is a novel and interesting transformation that might be useful in the appropriate setting. Nevertheless, we should stress that all the results in this chapter involving the BMS transformations generated by T (f) and R (Y ) are not aected by any issue that might arise regarding D (g). Readers that do not like superdilations can simply ignore the analysis involving D (g). The quantum analysis of the transformation generated by the vectors in (4.2) starts in sec- tion 4.2, where we consider an arbitrary CFT in any of the conformally at space-times introduced 65 in section 4.1. We write the conserved charges in terms of the stress tensor operator T , that in the Minkowski case are given by T (f) = Z I + dS T T (f) ; R(Y ) = Z I + dS T R (Y ) ; D(g) = Z I + dS T D (g) : (4.3) We show how these operators can be mapped by a CRT transformation 4 to the charges dened in the past regionI . Using the conformal transformation they can also be mapped to the charges at H + dened for the CFT in the curved space-times. We nish this section using the results in [115] (see also [116]) to prove the operators in (4.3) satisfy the same algebra as the associated vectors (4.2). In section 4.3 we study the action of the quantum charges (4.3) on the Hilbert space of the CFT. Applying the transformations on the vacuum allows us to study the following states jfie iT (f) j0i ; jYie iR(Y ) j0i ; jgie iD(g) j0i ; (4.4) that are dened onI in Minkowski (or H in the curved space-times). Remarkably, using the algebra satised by the charges together with some other natural ingredients, we are able to compute several features of these states in full generality, summarized in table 4.5. Perhaps the stronger of these results is that the statesjfi andjgi are equivalent to the vacuumj0i for any function f and g. This follows from the achronal Averaged Null Energy Condition (ANEC), proven for arbitrary QFTs in the previous chapters. In the remaining of section 4.3 we study the algebra satised by the charges (4.3) and construct representations in the Hilbert space of the CFT, for three and four space-time dimensions. In both cases we are able to make concrete statements that provide further insight into the action of these asymptotic transformations on the Hilbert space. In section 4.4 we propose a holographic description of the states (4.4), in the context of the AdS d+1 /CFT d correspondence where the bulk is well described by semi-classical gravity. As a rst step we extend the boundary vectors (4.2) into the bulk Bulk vectors = T (f); R (Y ) [ D (g) : (4.5) While in principle there is an innite number of ways of doing so, we x them by imposing the following conditions: 4 CRT is a discrete transformation analogous to CPT, but instead of a complete spatial re ection ~ x!~ x, it involves only a single componentx1!x1. While in even space-time dimensions both CRT and CPT are symmetries of any QFT, CPT is not when the space-time dimensions is odd (see subsection 5.1 of [87]). 66 Boundary CFT d Semi-classical gravity dual j0i g AdS p p j p ie i b Q[p] j0i g ( p )e p (g AdS ) h p j b Q[ q ]j p i Q g (p) [ q ] Table 4.1: Summary of our proposal for the holographic description of the boundary states (4.4), that correspond toj p i on the rst column with p =T;R;D respectively. The boundary charge b Q in the rst column are written in (4.3), where we add a hat to remind ourselves it is an operator. The metricg ( p ) in the second column is obtained by acting on the pure AdS d+1 metric with the (nite) transformation generated by p . Q g (p) [ q ] corresponds to the Noether charge associated to the vector q computed in the metric g ( p ). 1. As we approach the boundary we must recover the boundary vectors p ! p where p =T;R;D. 2. For some particular values of the functions f 0 ;Y A 0 ;g 0 entering in the denition of p in (4.2), the vectors generate ordinary conformal isometries. When xing the functions in this way for p , we require the bulk vectors to generate exact isometries of the AdS d+1 bulk space-time. 3. The algebra satised by the bulk vectors p must be exactly the same as the one obtained for the boundary vectors p . These conditions allow us to completely x the bulk vectors according to (4.101) and (4.118), for a three and arbitrary dimensional boundary respectively. Our proposal is that the boundary states (4.4) are described by a bulk geometry obtained by acting on the pure AdS d+1 metric with the (nite) transformation generated by p . We denote the resulting metric as g ( p ) e p (g AdS ). The ordinary AdS/CFT dictionary then gives the usual mapping between boundary expectation values and gravitational bulk Noether charges. In table 4.1 we summarize this holographic proposal, that we put to test in section 4.4 by computing the resulting bulk metrics and Noether charges, the nal results shown in tables 4.7 and 4.8. Comparing with the boundary CFT computations of section 4.3 (given in table 4.5) we nd agreement for all the quantities, proving strong evidence in favor of our holographic description of the states in (4.4). The agreement between bulk and boundary computations is not entirely perfect, given that 67 two Noether charges associated to the superdilation bulk vector D (g) diverge. A divergent charge associated to an asymptotic transformation is usually a sign that the metric fall-o condition pre- served by the vector is too permissive, that is precisely what we previously noticed in the analysis of section 4.1 for the boundary vector D (g) (see tables 4.4 and 4.5). As a result, we do not interpret the disagreement between bulk and boundary computations as a failure of the holographic pre- scription, but as evidence that superdilation is not a well behaved asymptotic symmetry. It would be interesting to understand how this issue arises directly from the boundary CFT perspective. We end this chapter in section 4.5, where we discuss the implications of our work regarding quantum hairs of asymptotically at (near-)extremal black holes. Building on our computations and focusing on a CFT in AdS 2 S d2 , we argue it is possible to construct an innite family of zero energy quantum states on both the future and past horizons, and asymptotic regions. The states on these surfaces are not independent but related in a precise way by conformal and CRT symmetry. Several appendices include important technical results used throughout the chapter. 4.1 Asymptotic (conformal) Killing vectors In this section we construct and study asymptotic Killing and conformal Killing vectors in d- dimensional Minkowski and a number of conformally at space-times (see table 4.2). The algebra satised by these vectors includes the BMS as a sub-algebra (supertranslations and superrotations) together with a novel transformation that we call `superdilation'. 4.1.1 Conformally at space-times Let us start by considering the d-dimensional Minkowski metric written as ds 2 =du 2 + 2dud +d 2 d2 2 ; (4.6) where = 1=r2 R 0 and u = tr, with t and r the ordinary time and radial coordinates in Minkowski. In these coordinates the null surface = 0 corresponds to future null innityI + . An analogous coordinate system allows us to describe past null innity. The metric in the unit sphere, S d2 , can be parametrized in stereographic coordinates ~ y2R d2 as d 2 d2 = 4jd~ yj 2 (1 +j~ yj 2 ) 2 ; (4.7) 68 AdS 2 S d2 dS d RS d1 AdS d w 2 (x ) 1 2 (1 +u) 2 2 (1 +u)(1 +u 2 ) + (=2) 2 (1 +u 2 ) 2 2 sin 2 ( ) 2 Table 4.2: Dierent choices for the conformal factor w 2 (x ) in (4.9) that result in a variety of interesting space-times. withj~ yj = 0;1 corresponding to the Poles of the sphere. There are several interesting conformal transformations we can apply to the Minkowski metric, obtained by rewriting (4.6) as ds 2 =w 2 (x ) " 2 du 2 + 2dud +d 2 d2 2 w 2 (x ) # ; (4.8) and performing a Weyl rescaling that removes the conformal factor w 2 (x ), so that the resulting space-time is given by d s 2 = 2 du 2 + 2dud +d 2 d2 2 w 2 (x ) : (4.9) Taking the conformal factor as indicated in table 4.2, we obtain a variety of conformally at space- times. Since in each case the connection is not entirely obvious, let us comment on each case separately. When the conformal factor is given by w 2 () = 1= 2 we obtain AdS 2 S d2 , where the AdS 2 factor is in Poincar e coordinates. We can see this more clearly by going to global coordinates (;)2R (0;), dened according to 2= = tan + + ( 0 ) 2 + tan ( 0 ) 2 ; u = tan ( 0 ) 2 ; (4.10) where =. In these coordinates the rescaled metric d s 2 in (4.9) becomes d s 2 AdS 2 S d2 = 2 du 2 + 2dud +d 2 d2 = d 2 +d 2 sin 2 () +d 2 d2 ; (4.11) that we recognize as AdS 2 S d2 in global coordinates, with the two AdS 2 boundaries located at = 0;, see gure 4.1. The original coordinates (;u) do not cover the whole AdS 2 space-time but only its Poincar e patch Poincar e patch : ( 0 ) ; (4.12) which corresponds to the shaded blue region in the left diagram in gure 4.1. Depending on the value of the parameter 0 2 (0;) appearing in the change of coordinates (4.10), the coordinates 69 AdS Boundary AdS Boundary North Pole South Pole North Pole South Pole de Sitter dS past boundary dS future boundary Figure 4.1: Penrose diagrams associated to AdS 2 S d2 , de Sitter and the Lorentzian cylinder RS d1 . The coordinates (u;) only cover the shaded blue region in each case, where several constant trajectories are sketched in gray. Future null innity at = 0 in Minkowski is mapped to the future horizon H + indicated with a dashed green line. cover a dierent region of global AdS 2 . The transformation maps the Minkowski asymptotic null innityI + at = 0 to the future Poincar e horizon of AdS 2 S d2 at + = 0 . Next, we can analyze the de Sitter case, which corresponds to taking the adequate conformal factor indicated in table 4.2. Same as in the previous case, it is instructive to rewrite the metric in the coordinates (;) in (4.10) but in this case with 0 ==2, so that the rescaled metric (4.9) becomes d s 2 dS d = 2 du 2 + 2dud +d 2 d2 (1 +u) 2 = d 2 +d 2 + sin 2 ()d 2 d2 cos 2 () : (4.13) We recognize this as global de Sitter, with the space-like boundaries atjj ==2. The coordinates (;u) do not cover the whole space-time but only the at slicing of de Sitter, see gure 4.1. The future null innity of MinkowskiI + is mapped to the cosmological horizon H + at = 0. To obtain the Lorentzian cylinder RS d1 we perform the Weyl rescaling in table 4.2 and change to the coordinates in (4.10) with an arbitrary value of 0 , so that we nd d s 2 RS d1 = 2 du 2 + 2dud +d 2 d2 (1 +u)(1 +u 2 ) + (=2) 2 (1 +u 2 ) 2 =d 2 +d 2 + sin 2 ()d 2 d2 : (4.14) The region covered by the coordinates (u;) is indicated in the right diagram of gure 4.1, where we see that = 0 now corresponds to the horizon H + at + = 0 . Finally, to make the connection between Minkowski and AdS d clear, we must explain the mean- 70 ing of the conformal factor in the last column of table 4.2. The angle 2 [;] is obtained by writing the metric in the unit sphere S d2 in (4.7) as d 2 d2 =d 2 + cos 2 ( )d 2 d3 : (4.15) The ordinary spherical angle is obtained by shifting ! +=2. The rescaled metric (4.9) in these coordinates becomes d s 2 AdS d = 2 du 2 + 2dud +d 2 + cos 2 ( )d 2 d3 sin 2 ( ) ; (4.16) where now the range of is restricted to 2 (0;=2], with the AdS d boundary being at = 0. These coordinates do not cover the full space-time but only the Poincar e patch, with the Poincar e horizon being at = 0. See section 4.4 for a construction of these coordinates from the embedding description of AdS. 4.1.2 Three dimensions We now consider asymptotic Killing and conformal Killing vectors dened in three dimensional Minkowski at future null innity, = 0 in (4.6). Using the conformal relations explained in the previous subsection means the asymptotic transformations are also dened for the horizons H + in the conformally at space-times. Let us start by writing the exact Killing and conformal Killing vectors of Minkowski in the coordinates (u;;), where the periodic anglejj is dened as y() = tan(=2), so thatd 2 1 =d 2 . Conformal transformations of Minkowski in these coordinates are generated by the following vectors T (f) =f()@ u f 0 ()@ 2 f 00 ()@ ; R (Y ) =uY 0 ()@ u + Y ()uY 00 () @ + Y 0 ()uY 000 () @ ; S (h) =u 2 h()@ u u(2 +u)h 0 ()@ 2(1 +u)h() + (2 +u) 2 h 00 () @ ; D (g) =ug()@ u ug 0 ()@ g() +ug 00 () @ ; (4.17) that correspond to translations, Lorentz transformations, special conformal transformations, and dilation respectively, where the four functions of are xed according to f 0 () =a 0 +a 1 cos() +a 2 sin() ; Y 0 () =b 0 +b 1 cos() +b 2 sin() ; h 0 () =c 0 +c 1 cos() +c 2 sin() ; g 0 () =d 0 : (4.18) 71 L (g uu ) L (g u ) b L (g uu ) b L (g u ) b L ( g uu ) b L ( g u ) T (f) 0 O(1) 0 O(1) 0 O( 2 ) R (Y ) O(1) O(1) O(1) O(1) O( 2 ) O( 2 ) S (h) 0 O(1= 2 ) 0 O(1) D (g) O(1) O(1=) O( 2 ) O() Table 4.3: Non vanishing ordinary and conformal Lie derivatives of the metric g and g = g =w 2 (x ) when considering the vectors in (4.17) for arbitrary functions f(), Y (), h() and g(). Since the vectors T (f) and R (Y ) have vanishing divergence we can replace the ordinary Lie derivatives in the rst two columns by the conformal derivatives. This gives the ten independent transformations of the conformal group in three dimensions, SO(3; 2). Let us denote the ordinary and the \conformal" Lie derivatives of the metric as L (g ) =r +r ; b L (g ) =r +r 2 d (r)g ; (4.19) whered is the space-time dimension (d = 3 in this case). The vectors T (f) and R (Y ) ( S (h) and D (g)) have vanishing (conformal) Lie derivative when the functions are xed according to (4.18). Looking at the form (4.17), it is natural to consider the more general class of transformations generated by arbitrary functions, not necessarily those given in (4.18), and compute the associated (conformal) Lie derivatives. In this case, there are two components of the metric that do not vanish but instead satisfy the fall-o conditions in indicated in the rst four columns in table 4.3. In other words, table 4.3 gives the asymptotic boundary conditions at = 0 that are preserved by the asymptotic conformal Killing vectors (4.17) dened by arbitrary functions f(), Y (), h(), and g(). Note that the way in which we have written the vectors in (4.17) in order to obtain the conformal transformations is non-unique. For instance, using that Y 0 () =Y 000 () is satised by (4.18) or the fact that g 0 () = 0, we can nd other ways of writing extensions of such vectors. The reason we have chosen this precise way among others is that when we promote (4.18) to arbitrary functions, the vectors satisfy the simple fall-o conditions given in table 4.3. In particular, note the (conformal) Lie derivative of the metric componentsg ,g andg vanish exactly, corresponding to the components that are xed exactly when writing an arbitrary asymptotically at metric in the Bondi gauge. 72 Let us now consider the action of the vectors in (4.17) on the rescaled metric g =g =w 2 in (4.9) after the Weyl transformation. While conformal Killing vectors are preserved under conformal transformations, this is not the case for exact Killing vectors. This becomes clear by noting the divergence of a vector behaves in the following way under a Weyl transformation ( r) = (r) +d(@) ln(w) ; (4.20) where r is the covariant derivative with respect to the rescaled metric (4.9). Since both vectors T (f) and R (Y ) have vanishing divergence computed with respect to the Minkowski metric, we can trivially replace the ordinary Lie derivative by the conformal version (4.19). Therefore, with respect to the rescaled metric it makes sense to compute conformal Lie derivatives of all the vectors. Doing so, we nd the Minkowski fall of conditions (shown in the rst four columns of table 4.3) simply get rescaled by the conformal factor as 1=w 2 (). Given that all of the conformal factors we are considering in table 4.2 have the same scaling behavior for small , i.e. w 2 () 1= 2 , all the conformally mapped space-time satisfy the fall-o conditions given in the last two columns in table 4.3. It is worth mentioning that, depending in the context, we can also consider the ordinary Lie derivative of the vectors in (4.17) for the curved space-times. For instance, the near horizon symmetries studied in [109] can be recovered from this perspective by considering the AdS 2 S d2 case. The asymptotic Killing vectors evaluated at = 0 constructed in [109] that preserve some particular boundary conditions take the following form 5 1 =0 =T ()@ u ; 2 =0 =X()u@ u ; 3 =0 =Y ()@ : (4.21) Comparing with the vectors dened in this work at = 0 (see (4.22) below), we see they can be obtained from (4.21) by considering simple linear combinations. Let us now consider the algebra satised by the vectors (4.17) for arbitrary functions. Since the vectors are in general pretty complicated, it is useful to rst evaluate them at = 0, where they 5 See equations (67) and (85) in [109] for the boundary conditions and (68) and (86) for the vectors. 73 have the following simpler structure T (f) =0 =f()@ u ; R (Y ) =0 =Y 0 ()u@ u +Y ()@ ; S (h) =0 =h()u 2 @ u 2h 0 ()u@ 2 h() + 2h 00 () @ ; D (g) =0 =g()u@ u : (4.22) Note that the vector S (h) is the only one with non-vanishing component in the direction, meaning the associated transformation makes the = 0 surface uctuate transversely. This is somehow expected as special conformal transformations leave the origin of the space-time xed while shift the asymptotic region. This feature has the consequence that the associated algebra does not close. 6 The algebra closes if we consider the remaining vectors at = 0, so that we nd T (f 1 ); T (f 2 ) = 0 ; T (f); R (Y ) = T ( b f ) ; b f() =f()Y 0 ()f 0 ()Y () ; R (Y 1 ); R (Y 2 ) = R ( b Y ) ; b Y () =Y 1 ()Y 0 2 ()Y 0 1 ()Y 2 () ; D (g 1 ); D (g 2 ) = 0 ; T (f); D (g) = T ( b f ) ; b f() =g()f() ; R (Y ); D (g) = D (b g ) ; b g() =g 0 ()Y () : (4.23) The rst three relations in (4.23) give the BMS algebra, with T (f) and R (Y ) generating super- translations and superrotations respectively. This algebra is naturally extended by incorporating the vector D (g), that generates the novel transformation we call `superdilation'. Note that these vectors and their associated algebra are not only dened on Minkowski but for all the conformally at space-times. Before generalizing to higher dimensions let us highlight a feature of the vectors S (h) and D (g) in (4.17). As can be seen from the fourth column in table 4.3, the conformal Killing equation for g u behaves likeO(1= 2 ) andO(1=) respectively. This is an asymptotic behavior that is much more singular than the conditions satised by the ordinary BMS vectors T (f) and R (Y ). Therefore we should be careful when studying the transformations generated by S (h) and D (g) 6 If we try restrict to functions h() such that the vector S (h) =0 has no component in the direction, then we nd the function h() is not periodic in . It is possible that a closed algebra can be obtained by considering a modied version of the Lie brackets that takes into account the variation of the metric [98, 99]. This might also be useful to show the algebra (4.23) is satised by the vectors away from = 0. 74 as complications can arise when extracting physical quantities associated to these vectors. This issue arises for superdilations in the holographic analysis of section 4.4. 4.1.3 Arbitrary dimensions Let us now generalize the previous discussion to arbitrary space-time dimensions d, where the Minkowski metric is given in (4.6) with the unit sphereS d2 described in stereographic coordinates ~ y2R d2 (4.7). The vectors generating conformal transformations that generalize (4.17) are given by T (f) =f@ u (D A f)@ A 2 d 2 (D 2 f)@ ; R (Y ) = (DY ) d 2 u@ u + Y A u d 2 D A (DY ) @ A + d 2 (DY ) u d 2 D 2 (DY ) @ ; S (h) =hu 2 @ u u(2 +u)(D A h)@ A 2(1 +u)h + (2 +u) 2 d 2 (D 2 h) @ ; D (g) =gu@ u u(D A g)@ A g + u d 2 (D 2 g) @ ; (4.24) where D A is the covariant derivative on the unit sphere S d2 . The vectors generating ordinary conformal vectors are obtained by taking the functions according to f 0 (~ y ) =a 0 + d2 X B=1 a B y B j~ yj 2 + 1 +a d1 j~ yj 2 1 j~ yj 2 + 1 ; Y A (~ y ) =b 0 y A + d2 X B=1 n w A B y B +p B 2y B y A AB (j~ yj 2 + 1) + ~ p B 2y B y A AB (j~ yj 2 1) o ; h 0 (~ y ) =c 0 + d2 X B=1 c B y B j~ yj 2 + 1 +c d1 j~ yj 2 1 j~ yj 2 + 1 ; g 0 (~ y ) =d 0 ; (4.25) wherew A B =w B A . The functionsf(~ y ) andh(~ y ) gived independent transformations correspond- ing to space-time translations and special conformal transformations, whileg(~ y ) =d 0 is the dilation. Lorentz transformations generated by Y A (~ y ) are determined by the parameters b 0 ;w A B ;p B ; ~ p B , that give the appropriate number of independent transformations corresponding to SO(d 1; 1), namely dim [Y A ] = 1 + (d 2)(d 3) 2 + (d 2) + (d 2) = d(d 1) 2 : (4.26) 75 L (g uu ) L (g uA ) L (g AB ) b L (g uu ) b L (g uA ) b L (g AB ) b L ( g uu ) b L ( g uA ) b L ( g AB ) T (f) 0 O(1) O(1=) 0 O(1) O(1=) 0 O( 2 ) O() R (Y ) O(1) O(1) O(1= 2 ) O(1) O(1) O(1= 2 ) O( 2 ) O( 2 ) O(1) S (h) 0 O(1= 2 ) O(1= 2 ) 0 O(1) O(1) D (g) O(1) O(1=) O(1=) O( 2 ) O() O() Table 4.4: Non vanishing ordinary and conformal Lie derivatives of the metric g and g = g =w 2 (x ) when considering the vectors in (4.24) for arbitrary functions f(~ y ) and Y A (~ y ), while h(~ y ) and g(~ y ) are restricted to (4.27). Let us now consider vectors of the form (4.24) but dened with arbitrary functions, not nec- essarily given by (4.25). When doing this for the conformal vectors S (h) and D (g), we nd the conformal Lie derivatives have non-vanishing components that explicitly violate some of the Bondi gauge conditions. It is therefore convenient not to consider completely arbitrary function h(~ y ) and g(~ y ) but to restrict them to those with the following dependence g(~ y ) =g(y A =y 1 ) ; h(~ y ) =h(y A =y 1 ) : (4.27) These functions do not depend in an arbitrary way on all the (d2) components ofy A , but only on the (d 1) coordinates obtained as y A =y 1 (there is no special role played by y 1 , as we can change this by any other component). As an example, for d = 5 we have g(~ y ) = g(y 2 =y 1 ;y 3 =y 1 ). With this restriction, the conformal Lie derivative of the metric satisfy much nicer relations that do not violate the Bondi gauge conditions, given in the rst six columns of table 4.4. 7 If we consider the transformations in the curved space-times obtained through the Weyl rescaling in (4.9), the conformal Lie derivatives are replaced by the fall-o conditions in the last three columns in table 4.4 (we have used that all the conformal factors in table 4.2 scale as w 2 () 1= 2 ). Same as in the three dimensional case, we obtain the associated algebra by rst evaluating the 7 The Bondi gauge conditions are g = gA = @[ 2 det(gAB)] = 0. The rst two are satised by the fall-o conditions given in table 4.4, while we have not checked the third condition. However if we consider the Newman- Unti gauge condition [117, 118], the third condition is replaced by gu = 2= 2 , that is preserved by the vectors in (4.24). 76 vectors at = 0 T (f) =0 =f@ u ; R (Y ) =0 = (DY ) d 2 u@ u +Y A @ A ; S (h) =0 =hu 2 @ u 2u(D A h)@ A 2 h + 2 d 2 (D 2 h) @ ; D (g) =0 =gu@ u : (4.28) Since the vector S (h) contains a non-trivial component in the direction, the full algebra of these vectors does not close. However, if we consider the algebra of the remaining vectors, it does close and is given by T (f 1 ); T (f 2 ) = 0 ; T (f); R (Y ) = T ( b f ) ; b f = (DY ) d 2 fY A D A f ; R (Y 1 ); R (Y 2 ) = R ( b Y ) ; b Y A =Y B 1 D B Y A 2 Y B 2 D B Y A 1 ; D (g 1 ); D (g 2 ) = 0 ; T (f); D (g) = T ( b f ) ; b f =fg ; R (Y ); D (g) = D (b g ) ; b g =Y A D A g : (4.29) The rst three relations form a subalgebra that corresponds to the ordinary BMS algebra, obtained from supertranslations and superrotations generated by T (f) and R (Y ) respectively. The full algebra also includes the superdilation vector D (g). Same as in the three dimensional case, these vectors and their algebra are dened in Minkowski as well as in any of the conformally at space- times discussed in subsection 4.1.1. 4.2 CFT charges and algebra In this section, we consider an arbitrary CFT and construct the quantum charges associated to the asymptotic (conformal) Killing vectors in (4.24). We start in subsection 4.2.1 by showing how the charges for the CFT in the various conformally at space-times are not independent but related between themselves by conformal symmetry. In subsection 4.2.2 we use a discrete symmetry of the CFT to map the charges in the future regions (eitherI + or H + ) to the regions in the past (I or H ). We nish in subsection 4.2.3 where we use the results of [115] to show all these charges satisfy the same algebra as the associated vectors in (4.29). 77 4.2.1 Conformal transformation of the charges Let us consider an arbitrary CFT in d-dimensional Minkowski space-time. The charge associated to a vector can be written in terms of the stress tensor operator, T , as follows Q[] Z dS T ; (4.30) where is a Cauchy surface in Minkowski with surface element dS =dSn , with n the future directed unit normal. When constructing a conserved charge Q with being an exact conformal Killing vector, the Cauchy surface we choose to write the operator is unimportant, as dierent choices for result in the same operator. For this reason, the charge is often said to be a topolog- ical operator. In this case, however, we are interested in constructing the charges for asymptotic conformal Killing vectors, which do not necessarily share this property precisely because they do not generate exact symmetries of the theory. As a result, the Cauchy surface we use to write the operator becomes important and turns out to be part of the prescription. While the full expression of the vectors in (4.24) is quite complicated, we consider the simpler case = 0, so that the charges obtained from (4.28) take the form Q[ T ]T (f) = lim !0 1 d2 Z S d2 d (~ y )f(~ y )E(~ y ) ; Q[ R ]R(Y ) = lim !0 1 d2 Z S d2 d (~ y ) (DY ) d 2 K(~ y ) +Y A N A (~ y ) ; Q[ D ]D(g) = lim !0 1 d2 Z S d2 d (~ y )g(~ y )K(~ y ) ; (4.31) where d (~ y ) is the volume element of S d2 and where we have dened the following light-ray operators E(~ y ) Z +1 1 duT uu (u; = 0;~ y ) ; K(~ y ) Z +1 1 duuT uu (u; = 0;~ y ) ; (4.32) and N A (~ y ) Z +1 1 duT uA (u; = 0;~ y ) : (4.33) For the CFT dened in the curved conformally at space-times we can apply exactly the same procedure to write the charges. However, instead of writing the charges in these space-times from scratch, it is convenient to apply the conformal transformations and map the Minkowski charges 78 (4.31) to the other space-times. This has the advantage that the functions f(~ y ), Y A (~ y ) and g(~ y ) in Minkowski space-time happen to determine the corresponding charges in the other space-times. To apply the mapping we use that the stress tensor transforms under a conformal transformation in the following way UT U y = @ x @x @ x @x T h T i 0 w( x) d2 ; (4.34) where U is the unitary operator implementing the conformal transformation U :H! H, and we add a bar over quantities after the mapping. Using this on the general expression for the charge Q[] in (4.30), we nd Q[]UQ[]U y = Z H + d S T h T i 0 ; (4.35) where both vectors n and are now written in the new coordinates x and we have dened d S dSn =w d2 . For each of the dierent conformal factors given in table 4.2 the surface element can be written as d S = u d (~ y )du 8 > > > > > < > > > > > : 1 ; AdS 2 S d2 and dS d ; (1 +u 2 ) 2d 2 ; RS d1 ; sin( ) 2d ; AdS d : (4.36) The termh T i 0 in (4.35) corresponds to the vacuum expectation value and appears due to the anomalous transformation of the stress tensor (ford = 2, it is xed by the Schwartzian derivative). To write the mapped charges in (4.35) we must compute the componentsh T uu i 0 andh T uA i 0 at = 0, that are highly constrained by symmetry. Consider the vacuum state of a QFT dened on a geometryM obtained as the product of two maximally symmetric manifoldsM =M 1 M 2 . Using the isometries in each factor and the fact that the vacuum statej0i remains invariant, we can reduceh T i 0 to (see appendix A.1) h T i 0 =a 1 g (1) ij +a 2 g (2) ab ; (4.37) where (a 1 ;a 2 ) are constants and ( g (1) ij ; g (2) ab ) are the metrics in each maximally symmetric manifold. 8 We can use this to compute the vacuum contribution of the stress tensor in (4.35). Since (4.37) 8 Since the coordinates (u;;~ y ) we are using to describe the curved manifold do not cover the whole space-time (see blue regions in gure 4.1), one might ask whether the vacuum state appearing in the expectation value in (4.35) is the same as the global vacuum in (4.37). For instance, this is the case when comparing the usual Minkowski and Rindler vacuum, which are distinct states. The global vacuum is most conveniently dened by analytically continuing 79 applies for the metric written in terms of the global coordinates (;) in (4.10), we must rst compute the appropriate components in the global coordinates and then translate to (u;), so that the nal result gives h T uu i 0 = 8 < : 0 ; for AdS 2 S d2 ; dS d ; AdS d ; 6= 0 ; for RS d1 ; h T uA i 0 = 0 : (4.38) The componenth T uA i 0 vanishes since all the metrics have zero non-diagonal components g A and g A . The contributions toh T uu i 0 vanish when they are obtained by projecting along a null tangent vector. This is not the case for the CFT on the Lorentzian cylinder since the null geodesic moving alongH + at ( + ;~ y ) = ( 0 ;~ y 0 ) in the global coordinates (4.10) has a tangent vector with non-trivial components alongR and S d1 which does not vanish because the constants (a 1 ;a 2 ) in (4.37) are dierent. This non-zero component can be written explicitly from (2.23). We can now use (4.36) and (4.38) to write the mapped charge in (4.35) explicitly. For the AdS 2 S d2 and dS d cases the nal result is the same as in (4.31) but without the prefactor 1= d2 . Note that the coordinate u is an ane parameter for the null geodesics in Minkowski, dened as = 0 with constant~ y . This follows from checking that the geodesic equation is satised in the Minkowski metric (4.6) d 2 x du 2 + dx du dx du = 0 : (4.39) If u was not ane, then the right-hand side of (4.39) would be proportional to the tangent vector along the curve. Having an ane parameteru is important since it allows us to identify the light-ray operatorE(~ y ) in (4.32) as the ANEC, meaningE(~ y ) 0. When we apply a conformal transformation, the geodesic equation (4.39) is not invariant since the connection transforms with an anomalous term due to the Weyl rescaling. Same as in the discussion around (2.9), the coordinate u remains ane only if the following condition is satised u is ane in g () dw(x) du =0 = 0 : (4.40) the Lorentzian timet totE =it and considering the path integral over half of the Euclidean manifoldtE < 0, namely j0i = R t ;t E <0 D exp (IE []), where are the elds of the QFT and IE [] the Euclidean action. From this description of the vacuum we see that as long as there is a Cauchy surface that ts in the region covered by the coordinates, the vacuum state will be equivalent to the global vacuum. From the diagrams in gure 4.1 we see we can always choose a Cauchy surface of the whole manifold that ts entirely in the blue region. For the Rindler region in Minkowski this is not the case, and therefore the vacuum states dened in each case are dierent. While this is a formal argument, it can be shown explicitly for a free scalar by comparing the vacuum two-point functions in each quantization scheme. See [2] for AdS2 and section 3 of [119] for dS d . 80 From the rst line in (4.36) we see this is satised in all space-times except for the Lorentzian cylinder. In this case, we can dene a new parameter = (u) that is ane inRS d1 (i.e. it satises the geodesic equation as written in (4.39)) according to 9 u() = tan() ; jj=2 : (4.41) The charges inRS d1 are then given by (4.31) without the factor 1= d2 and with the following light-ray operators E(~ y ) = Z =2 =2 d cos d () T h T i 0 ; K(~ y ) = Z =2 =2 d sin() cos d1 () T h T i 0 ; N A (~ y ) = Z =2 =2 d cos d2 () T A ; (4.42) where E(~ y ) is the positive operator studied in chapter 2, and we have also included the non-vanishing vacuum contribution from (4.38). The mapping to AdS d is slightly more involved since in that case we must not only consider a Weyl rescaling of the metric but also a change of coordinates in the sphere S d2 , according to (4.15). This requires us to change the angular coordinates of the vectors generating the transformations (4.28). While this is certainly a straightforward computation, we shall only write the three dimensional case explicitly, where the vectors (4.22) are already written in terms of the angle. The Weyl rescaling in this case is given byw 2 (;) = sin 2 ()= 2 , so that the AdS 3 metric (4.15) is d s 2 AdS 3 = 2 du 2 + 2dud +d 2 sin 2 () ; (4.43) where the range of in this case is given by 2 (0;), with ! 0; corresponding to dierent ways of approaching the same AdS 3 boundary. The charges can be then written from (4.35), (4.36) and (4.38) as T (f) = Z 0 d sin() f() Z +1 1 du T uu ; R(Y ) = Z 0 d sin() Z +1 1 du Y 0 ()u T uu +Y () T u ; D(g) = Z 0 d sin() g() Z +1 1 duu T uu : (4.44) 9 The ane parameter here is related to in (2.40). 81 The takeaway from this subsection is that given the charges (4.31) dened inI + and determined by the functions f(~ y );Y A (~ y );g(~ y ) , the charges inH + are not written in terms of new functions, but the same ones as in the Minkowski. 4.2.2 Discrete transformation between future and past regions Our analysis so far applies to the regions in the futureI + for Minkowski and the horizon H + for the curved space-times (see gure 4.1). A completely analogous analysis can be performed for the corresponding asymptotic surfaces located in the past regions. In this subsection we show the charges in the future and past regions are not independent but related in a very precise and interesting way through a discrete symmetry of the CFT. Given a QFT in the space-time g the vacuum is invariant under the action generated by the charges associated to the isometries of the space-time that are smoothly connected to the identity. Generically, discrete isometries such as a time reversal in Minkowski, are not symmetries of the ground state. However, there are certain combinations of discrete symmetries that leave the vacuum invariant. In the case of Minkowski, a transformation that leaves the vacuum invariant is given by the following transformation 10 CRT : (t;x 1 ;~ x )! (t;x 1 ;~ x ) ; (4.45) where (t;x 1 ;~ x ) are ordinary Cartesian coordinates, ~ x = (x 2 ;:::;x d1 ). We can use this symmetry to relate the charges in the future and past regions. Instead of considering the Minkowski case, let us focus on the more interesting setup of a CFT on AdS 2 S d2 , where the future Poincar e horizon H + we have been considering so far is located at + = 0 2 [0;] in terms of the global coordinates in (4.11). We dene the past horizon H as the surface = 0 , which can be conveniently described in terms of a new set of coordinates (v;%) dened similarly to (4.10) as 2=% = tan + ( 0 ) 2 + tan + ( 0 ) 2 ; v = tan + ( 0 ) 2 : (4.46) These coordinates only cover a Poincar e patch of AdS 2 given by Poincar e patch : ( 0 ) ; (4.47) 10 While in even space-time dimensions the CRT symmetry is equivalent to the more standard CPT transformation, for odd dimensions the CPT is not a symmetry of the QFT, see subsection 5.1 in [87] 82 AdS Boundary AdS Boundary AdS Boundary AdS Boundary Time reection Figure 4.2: On the left we have the region of AdS 2 covered by the coordinates (v;%) dened in (4.46). The past horizon (marked in red) is obtained as % = 0 andv2R. On the right we plot the region of AdS 2 covered by the coordinates (u;) and (v;%) in green and red respectively. The CRT transformation in (4.52) gives a map between these regions, most importantly mapping between the horizons H + $H . and plotted in the left diagram of gure 4.2. Note that for general 0 2 [0;] this is a dierent Poincar e patch that the one covered by the coordinates (u;) in (4.12), shown in green on the right diagram of the same gure. It is only for 0 = that both set of coordinates cover the same region. The AdS 2 S d2 metric (4.11) in these coordinates becomes d s 2 AdS 2 S d2 =% 2 dv 2 2dvd% + 4d~ zd~ z (1 +j~ zj) 2 = d 2 +d 2 sin 2 () +d 2 d2 ; (4.48) where we have taken stereographic coordinates ~ z2R d2 to describe the sphere S d2 . The past horizon H at = 0 is located at % = 0. It is easy to check the asymptotic (conformal) Killing vectors on the past horizon H are completely analogous to those given in (4.24) after replacing (u;;~ y )! (v;%;~ z ). The charges 83 constructed from these vectors evaluated at % = 0 are similar to (4.31) and given by 11 Q[ T ] T (f ) = Z S d2 d (~ z )f (~ z ) E(~ z ) ; Q[ R ] R (Y ) = Z S d2 d (~ z ) (DY ) d 2 K(~ z ) +Y A N A (~ z ) ; Q[ D ] D (g ) = Z S d2 d (~ z )g (~ z ) K(~ z ) ; (4.49) where we have added a minus subscript on the charges and functions to distinguish these from those associated to the future regions, which should now contain a plus subscript. The light-ray operators in (4.49) are analogous to those dened previously, namely E(~ z ) Z +1 1 dv T vv (v;% = 0;~ z ) ; K(~ z ) Z +1 1 dvv T vv (v;% = 0;~ z ) ; (4.50) and N A (~ z ) Z +1 1 dv T vA (v;% = 0;~ z ) : (4.51) Naively, it might seem the charges in each of the horizons are unrelated, meaning the functions f ,Y A andg are completely independent. However, this is not the case since an analogous CRT symmetry in AdS 2 S d2 relates these functions in a precise way. In appendix A.3 we show any QFT in AdS 2 S d2 is invariant under the following discrete transformation CRT : (;;~ y ) ! ;;~ z = ~ y j~ yj 2 ; (4.52) realizing (CRT)j 0i =j 0i in this case. Apart from the time re ection we have an inversion in the stereographic coordinates on the sphereS d2 . 12 The usefulness of this transformations comes from the fact that it maps between the future horizon H + at + = 0 to H at = 0 (see right diagram in gure 4.2). Applying the adjoint action of the CRT operator on the stress tensor we nd 13 (CRT) T (x)(CRT) 1 = @~ x @x @~ x @x T (~ x) ; 11 The charges in (4.49) do not have the prefactor 1= d2 in (4.31) since the AdS2S d2 surface element is given by (4.36). After making the replacement (u;;~ y )! (v;%;~ z ) on the vectors (4.28) at% = 0 we see the supertranslation vector picks up a minus sign, that appears in the supertranslation charge in (4.49). 12 This discrete transformation is completely analogous to the one in Minkowski (4.45). Taking spherical coordinates in Minkowski (t;r;~ y ), with~ y parametrizing theS d2 , one can check the transformation (4.45) is equivalent to (4.52) replacing !t, see appendix A.3 for details. 13 While the charge conjugation operator C implements a hermitian conjugate it makes no dierence in this case since the stress tensor is Hermitian. 84 where ~ x are the transformed coordinate. Using this we can map the charge on H + to H in the following way (CRT) Q + (CRT) 1 = (CRT) Z H + dSn T (CRT) 1 = Z H dSn T ; (4.53) where the vectors n and are the ones dening Q + but written in the transformed coordinates in (4.52). Writing the nal expression in terms of the (v;%;~ z ) coordinates we nd (CRT) T + (f + )(CRT) 1 = T (f ) ; where f (~ z ) =f + (~ z=j~ zj 2 ) ; (CRT) R + (Y + )(CRT) 1 = R (Y ) ; where Y A (~ z ) = @z A @y B Y B + (~ z=j~ zj 2 ) ; (CRT) D + (g + )(CRT) 1 = D (g ) ; where g (~ z ) =g + (~ z=j~ zj 2 ) ; (4.54) where the charges in the past region are given in (4.49). The minus sign appearing in the relation between the functions comes from the fact that the normal vector n = u in the coordinates on H is n = v . This shows the charges in each horizon are not independent but related in an interesting way. Through the conformal map explained in the previous subsection, this analysis together with the relations in (4.54) are not exclusive to the CFT in AdS 2 S d2 but to any of the conformally at space-times. Using the invariance of the vacuum under CRT transformation also allows us to map states, for instance (CRT)jY + i (CRT)e i R(Y + ) j 0i = (CRT)e i R(Y + ) (CRT)j 0i =e i R(Y ) j 0i =jY i : (4.55) In section 4.3, we study features of states such asjY + i in detail. 4.2.3 Charge algebra We nish this section by considering the algebra of the charges associated to the asymptotic (con- formal) Killing vectors. Since the charges in the dierent setups are related through the adjoint action ofU in (4.35) or (CRT) in (4.53), which preserves the structure of any algebra, we know from the beginning that the algebra satised by the charges in Minkowski or any of the conformally at space-times must be exactly the same. Since the asymptotic vectors at = 0 in (4.28) satisfy the algebra (4.29), the expectation is that the charges realize the same algebra, which can be written 85 as T (f 1 );T (f 2 ) = 0 ; T (f);R(Y ) =iT ( b f ) ; b f = (DY ) d 2 fY A D A f ; R(Y 1 );R(Y 2 ) =iR( b Y ) ; b Y =Y B 1 D B Y A 2 Y B 2 D B Y A 1 ; D(g 1 );D(g 2 ) = 0 ; T (f);D(g) =iT ( b f ) ; b f =fg ; R(Y );D(g) =iD(b g ) ; b g =Y A D A g ; (4.56) where the factor i arises since the charges are Hermitian operators. Showing this operator algebra is satised is highly non-trivial, since it involves computing complicated commutators involving the stress tensor. We can still show (4.56) is satised by building on the results of [115], where the algebra of light-ray operators on the Minkowski null plane where computed for arbitrary CFTs. In appendix A.2 we apply a conformal transformation from chapter 2 to the algebra of light-ray operators of [115] from the Minkowski null plane to AdS 2 S d2 and use it to show (4.56) is indeed satised. 4.3 Quantum transformations on the Hilbert space In previous sections we have considered asymptotic (conformal) Killing vectors in several confor- mally at space-times, constructed the quantum charges and shown they satisfy the extended BMS algebra in (4.56). In this section, we study the action of these transformations on the Hilbert space, implemented through the chargesT (f),R(Y ) andD(g), corresponding to supertransla- tions, superrotations, and superdilations respectively. For concreteness, here we focus on the CFT in Minkowski with the charges dened at future null innityI + , even though the results apply in much more generality. In subsection 4.3.1, we consider the group action obtained by exponenti- ating the charges on the vacuum statej0i of the CFT. In subsection 4.3.2, we study vacuum and non-vacuum representations of the algebra (4.56) on the Hilbert space of the CFT, where we focus on the three and four dimensional cases. 86 4.3.1 Group action Let us start by recalling some basic notions of symmetries and conserved charges in CFTs. For any vector , there is an associated conserved charge Q[] that can be written in terms of the stress tensor as in (4.30). The nite transformation is then implemented on the statesj i2H by exponentiating the charges in the standard wayj s i = e isQ j i, with s2R. For CFTs, if the vector satises the Killing or conformal Killing equation, the vacuum state is invariant under the transformation, which impliesj0i is an eigenstate ofQ 0 . In this case, the charge annihilates the vacuum Q 0 j0i = 0 since the vacuum expectation value of the stress tensor vanishesh0jT j0i = 0. The charges that annihilate the vacuum are given in (4.31) Q 0 = n T (f 0 );R(Y 0 );D(g 0 ) o ; (4.57) with the functions f 0 (~ y ), Y A 0 (~ y ) and g 0 (~ y ) xed by (4.25). In this chapter we are considering more general charges that are associated to asymptotic (con- formal) Killing vectors, obtained by taking arbitrary functions not necessarily given by (4.25). This means the charges are not required to annihilate the vacuum but instead induce some non-trivial transformation. Applying the nite transformations on the vacuum allows we can dene jfie iT (f) j0i ; jYie iR(Y ) j0i ; jgie iD(g) j0i : (4.58) Since the charges are evaluated at = 0, these states are dened on this surface. To characterize them we rst analyze the action of the light-ray operators (4.32) and (4.33) on the vacuumj0i. ForE(~ y ) we can use that the achronal ANEC implies 14 k p E(~ y )j0ik 2 =h0jE(~ y )j0i = 0 =) E(~ y )j0i = 0 : (4.59) Although the operatorK(~ y ) is not positive denite, it also annihilates the vacuum. To see this, we use some results of [120] where it was shown that the chargeQ + associated to a particular global conformal transformation in the null direction satises the following commutation relation 15 E(~ y );Q + = 4iK(~ y ) : (4.60) 14 The corresponding operator E(~ y ) in the Lorentzian cylinder (4.42) is not the achronal ANEC but still a positive operator, e.g. as shown in (2.22). 15 This is obtained from the third line in equation (3.9) of [120] with n =2. Although that work is considering the light-ray operators in the null plane instead of null innity, they are related by a conformal transformation, see appendix A.2. 87 j i T (f 0 )j iR(Y 0 )j iD(g 0 )j i T (f)j iR(Y )j iD(g)j i hT (f)i hR(Y )i hD(g)i j0i 0 0 0 0 6= 0 0 0 0 0 jYi 0 6= 0 0 0 6= 0 0 0 0 0 Table 4.5: Action of the charges (4.31) on the vacuum statej0i and the statejYi =e iR(Y ) j0i. The rst three columns correspond to the charges (4.57) that generate ordinary conformal trans- formations. By \6= 0" we mean the state is not an eigenstate of the charge, but it has a non-trivial action. Using thatQ + j0i =E(~ y )j0i = 0 we seeK(~ y ) also annihilates the vacuum. For the remaining light-ray operatorN A (~ y ) there is no analogous argument, meaning it generates some non-trivial action onj0i. Since both the supertranslation and superdilation charges are obtained by integratingE(~ y ) and K(~ y ) over the sphere S d2 (4.31), we haveT (f)j0i =D(g)j0i = 0 for any two functionsf(~ y ) and g(~ y ), meaning nite supertranslations and superdilations act trivially on the vacuum state jfi =e iT (f) j0i = 1 X n=0 (i) n n! T n (f)j0i =j0i ; jgi =e iD(g) j0i = 1 X n=0 (i) n n! D n (g)j0i =j0i : (4.61) This is a very general and strong result. Although f(~ y )@ u and g(~ y )u@ u are not exact Killing vectors, the vacuum state is still invariant under the associated transformation. This is certainly not the case for the superrotation chargeR(Y ) which is built from the light-ray operatorN A (~ y ) (4.31). Notice, however, that this light-ray operator still has vanishing expectation value since it is built from integrating the stress tensor T . These results are summarized in the rst row of table 4.5. It is important to stress the equivalence between the statesj0i =jfi =jgi holds for = 0, they are distinct states away from the null surface. We can then study features of the statesjYi in (4.58). To do so, we use the following two algebraic identities that hold for arbitrary operators V and W We iV j0i = " e iV W + 1 X n=1 n X m=1 n m (i) n (1) m n! V nm L m V (W ) # j0i ; h0je iV We iV j0i = 1 X n=0 i n n! h0jL n V (W )j0i ; (4.62) whereL V (W ) = [V;W ],L 2 V (W ) = [V; [V;W ]] and so on. While the second identity is standard, we 88 prove the rst one in appendix A.4. Using these relations, together with the algebra (4.56) satised by the charges and the transformation properties of the vacuumj0i given in the rst row in table 4.5, we obtain the second row. Let us now comment on the most salient features of this table. In the rst three columns we have the action of the charges associated to ordinary conformal transformations on the statejYi. One of the charges inT (f 0 ) is obtained from f 0 (~ y ) = 1, which corresponds to rigid translations in the u time-like coordinate of the Minkowski metric (4.6), meaning that H u T (f 0 = 1) is the associated Hamiltonian. Hence, the rst column in table 4.5 implies H u j0i =H u jYi = 0 : (4.63) This showsjYi gives an innite number of eigenstates of the Hamiltonian H u with minimum eigenvalue and thus can be regarded as soft modes. While the states are also invariant under rigid dilations generated byD(g 0 ), Lorentz transformations induce non-trivial transformations. This means there is not a true innite degeneracy of the vacuum statej0i, since we can distinguish it from the statesjYi by applying ordinary Lorentz transformations. From the fourth to the sixth columns in table 4.5, we see that all the states are invariant under arbitrary supertranslations and superdilations, same as the vacuum statej0i. This follows from the ANECE(~ y )j0i =K(~ y )j0i = 0 together with the rst identity in (4.62), where the second term is computed using the algebra (4.56). On the other hand, we see that all the states have a non-trivial transformation under superrotations. However, using the second identity in (4.62) together with the algebra, we can show the vacuum expectation of the charges in all these states vanish, as indicated in the last three columns in table 4.5. All the results given in table 4.5 apply to arbitrary CFTs in Minkowski space-time as well as on any of the conformally related space-times dened by performing the Weyl rescaling in (4.9). While one can study the action of the nite transformations on non-vacuum states, the setup becomes more complicated and it is dicult to make concrete statements. 4.3.2 Algebra representations We can get a more detailed characterization of the action of the charges on the Hilbert space by studying the algebra representations of (4.56). Since the algebra greatly depends on the space-time 89 dimension, we focus on the three and four dimensional cases. Three dimensions For the three dimensional case the vectors generating the transformations at = 0 (4.22) only depend on the angle . To study its algebra, it is convenient to expand the functions f(), Y () and g() in a Fourier series f() = X n2Z f n e in ; Y () = X n2Z Y n e in ; g() = X n2Z g n e in ; (4.64) where real functions demand the coecient expansion satisfy c n =c n . Using this in the denition of the charges in (4.31) we nd T (f) = X n2Z f n lim !0 Z S 1 d e in E() X n2Z f n T n ; R(Y ) = X n2Z Y n lim !0 Z S 1 d e in (inK() +N A ()) X n2Z Y n R n ; D(g) = X n2Z g n lim !0 Z S 1 d e in K() X n2Z g n D n : (4.65) Since the charges are Hermitian we have that all the mode operators verify P y n =P n . Using this expansion on the algebra in (4.56) we nd [T n ;T m ] = [D n ;D m ] = 0 ; [T n ;R m ] = (nm)T m+n ; [R n ;R m ] = (nm)R n+m ; [T n ;D m ] =iT n+m ; [R n ;D m ] =mD n+m : (4.66) From these relations we identify several interesting sub-algebras. The subsetsfT n g,fD n g andfR n g give two abelian and a Witt sub-algebra respectively, whilefT n ;R m g corresponds to the standard BMS 3 algebra, obtained as a semi-direct sum of the abelian and Witt sub-algebras [121]. The subsetfD n g;fR m g is an algebra that has appeared in other contexts, like in [109]. The subset fD n g;fT m g can be regarded as an innite-dimensional extension of the Borel subalgebra of sl(2). In the following we study vacuum and non-vacuum representations of this algebra (see [122] for related work on the BMS 3 algebra). 90 Vacuum representation: The starting point for the vacuum representation is its invariance under the action generated by the following modes T n j0i =D n j0i = 0 ; n2Z ; R n j0i = 0 ; n = 0;1 : (4.67) While the rst line is implied by the ANEC, the second is a consequence of the invariance of the vacuum under ordinary Lorentz transformations, that correspond to taking Y 0 () in (4.18). The vector space of the representation is spanned by acting successively with the operators R n on the vacuum statej0i, which gives jfm j gi k Y j=1 R m j j0i ; m j m j+1 : (4.68) States with dierent ordering can be put in this form using the algebra (4.66). From the conditions in (4.67) and the algebra we can use induction to prove these states satisfy the following properties T n jfm j gi =D n jfm j gi = 0 ; R 0 jfm j gi =M jfm j gi ; M k X j=1 m j : (4.69) We see that all supertranslations T n and superdilations D n annihilate all the states in the vacuum representation, in agreement with the previous analysis in general dimensions, summarized in ta- ble 4.5. Since the hermitian operatorR 0 is the angular momentum generator in the direction, the statesjfn i g;fm j gi are angular momentum eigenstates with integer eigenvalue M in (4.69), that gets contributions both from theR m modes. For a xed eigenvalueM there is a huge degeneracy, since there is a (very large) innite number of ways to obtain M by summing integer numbers. It is instructive to compare with highest weight representations of the Virasoro algebra. In that setup, the operator playing a similar role to the angular momentum R 0 is the energy L 0 . Since a well dened CFT requires the energy to be bounded from below, we get that half of the operators in the Virasoro algebra L n annihilate the vacuum. For the representation in (4.69) the situation is very dierent, as there is no reason for the angular momentum to have a bounded spectrum, given that a state can have arbitrary angular momentum in either direction. This is re ected in the fact that the eigenvalue M in (4.69) gets non-trivial contributions for integer values of m j . While the angular momentum eigenstatesjfm j gi provide a clear picture for the vector space of the vacuum representation, it is not the whole story since we cannot compute their inner product. 91 The algebra together with the conditions in (4.67) are not enough to do so. However, we can proceed similarly as in [122] and construct states with well dened norm by instead considering the states H = n jYih0j0i 1=2 e iR(Y ) j0i : Y ()2 Di(S 1 ) o ; (4.70) whereR(Y ) is the full charge given in (4.65). The crucial feature ofH is that all of these states have unit norm by construction. 16 Although we are exponentiating the charges as we do when considering the group representation, we should think of (4.70) as a particular change of basis from the angular momentum eigenstates in (4.68). Before moving on to consider non-vacuum representations, let us write a simple element of the Hilbert space in (4.70) explicitly, obtained by taking Y () = 2" cos(k) for some integer k, so that we get j0; 2" cos(k)i =e i"(R k +R k ) j0i =j0ii"(R k +R k )j0i +O(" 2 ) : (4.71) From this expression we see that the hermiticity condition on the charges forces that any partic- ular mode R k must be accompanied by a mode with the same magnitude but inverse direction. Every time angular momentum is added in one direction of the circle, another excitation must be considered in the opposite direction. From this perspective it is reasonable to obtain a vanishing expectation value of the angular momentumhYjR 0 jYi = 0 when taking the average, as previously obtained in table 4.5. Non-vacuum representations: We now consider non-vacuum representations of the algebra (4.66), where we start from an excited statej i that is an eigenstate of a commuting subset of the operators that generate the ordinary conformal transformations Q 0 =fT 0 ;T 1 ;R 0 ;R 1 ;D 0 g : (4.72) There are three commuting subsets of operators we can chose from S 1 =fT 0 ;T 1 g ; S 2 =fT 0 ;R 0 g ; S 3 =fR 0 ;D 0 g : (4.73) 16 While the statesjYi have well dened norm we would like to have states that are orthonormal, i.e. hY 0 jYi = (Y 0 ;Y ), where (;) is a Dirac delta dened with respect to an appropriate measure in the space of functions. For a similar setup in [122] it was argued in favor of the existence of such measure and the irreducibility of the representation. 92 We nd S 2 to be the most convenient, so that thej i corresponds to an energy and angular momentum eigenstate 17 T 0 j i =E u j i ; R 0 j i = j i ; (4.74) whereE u > 0 and 2Z. A dierence with respect to the vacuum representation is that the state j i is not invariant under arbitrary supertranslations T n and superdilationsD n , ash jE()j i6= 0 so that the argument in (4.59) does not apply. States in this representation are obtained by acting onj i with any element of the algebra jfp r g;fn i g;fm j g; i s Y r=1 T pr ` Y i=1 D n i k Y j=1 R m j j i ; m j m j+1 : (4.75) Acting on these states with the Hamiltonian T 0 and the angular momentum R 0 we can use the algebra (4.66) to show R 0 jfp r g;fn i g;fm j g; i =M jfp r g;fn i g;fm j g; i ; M s X r=1 p r ` X i=1 n i k X j=1 m j (4.76) together with a cumbersome expression for T 0 jfp r g;fn i g;fm j g; i that does not give an energy eigenstate. On the other hand, all the states in (4.75) are angular momentum eigenstates, where the action of all the elements in the algebra contribute to the eigenvalueM . This gives a generalization of the second relation in (4.69) for the vacuum representation. Similarly, while we cannot compute the norm of the states in (4.75), we can consider the normalized states constructed as H = n jf;g;Yih j i 1=2 e i(T (f)+D(g)+R(Y )) j i : f();g();Y ()2 Di(S 1 ) o ; (4.77) where we are assuming the statej i the representation is built from is normalizable (see footnote 17 for an example in which this is not the case). 17 One can also study representations built from S1, that can be extendedfTng for all n2Z, although this does not seem very useful. The subset S3 is also interesting, in which R0j i = j i and D0j i = j i. In Lorentzian signature the eigenvalue ofD0 does not correspond to the discrete and positive scaling dimensions of the operators in the CFT. We can see this by considering the stateT0j i and using the algebra (4.66) to show it is also an eigenstate of D0 but with complex eigenvalue i. This is consistent with D0 being hermitian only if the statej i is not normalizable. Exactly this same issue arises when considering representations of the ordinary conformal algebra, starting from the statejOiO(0)j0i whereO(x ) is a primary hermitian operator. Using the Ward identities and the conformal algebra one nds that while the statejOi has the same issues as explained forj i, it is not normalizable sincehOjOi =h0jO(0) 2 j0i = undened. 93 Four dimensions Let us now study representations of the algebra (4.56) in the four dimensional case, where it is convenient to use the complex coordinate z =y 1 +iy 2 to parametrize the unit two-sphere, so that the metric (4.7) becomes d 2 2 = 4dzd z (1 +z z) 2 : (4.78) The arbitrary functions f(z; z) and g(z; z) on S 2 that dene the asymptotic (conformal) Killing vectors in (4.28) can be expanded in a Laurent expansion as f(z; z) = 1 1 +jzj 2 X n;m2Z f (n;m) z n z m ; g(z; z) = X n;m2Z g (n;m) z n z m : (4.79) We have added the factor (1 +jz 2 j) 1 to the expansion off(z; z) (as in [99]) so that we can recover the functions in (4.25) that result in rigid space-time translations by considering a nite number of modes. Demanding the functions are real constraints the coecients to satisfy f (n;m) =f (m;n) and g (n;m) =g (m;n) . For the vector Y A (z; z) we should in principle apply a similar procedure, in which we expand the two components in a Laurent series in both z and z. However, as a rst step we constraint ourselves to a holomorphic ansatz given by Y z (z; z) =Y (z) ; Y z (z; z) = Y ( z) : (4.80) This is the usual approach taken in the BMS literature when instead of considering `super-Lorentz' transformations (obtained by Di(S 2 )) one considers superrotations by restricting to (4.80). The six rigid Lorentz transformations given in (4.25) in terms of the stereographic coordinates ~ y, are obtained in complex coordinates by takingY (z) =a+bz +cz 2 witha,b andc complex parameters. The holomorphic function Y (z) can be expanded in a Laurent series as Y (z) =i X m2Z Y m z m+1 : (4.81) We can now use the expansions in (4.79) and (4.81) in the Minkowski charges (4.31) at null 94 innity to dene the following mode operators T (f) = X n;m2Z f (n;m) lim !0 Z S 2 d 2 2 z n z m 1 +jzj 2 E(z; z) X n;m2Z f (n;m) T (n;m) ; R(Y ) = X m2Z Y m i lim !0 Z S 2 d 2 2 m + 1 2 jzj 2 1 +jzj 2 z m K(z; z) +z m+1 N z (z; z) + h:c: = X m2Z h Y m R m + Y m R y m i ; D(g) = X n;m2Z g (n;m) lim !0 Z S 2 d 2 2 z n z m K(z; z) X n;m2Z g (n;m) D (n;m) ; (4.82) where the additional term in the superrotation modes R m comes from the connection on S 2 when computing the divergence of Y A (z; z). Since the full charges are hermitian operators, we get the following conditions on the modes T y (n;m) =T (m;n) ; D y (n;m) =D (m;n) ; (4.83) whileR m andR y m are independent. Using the algebra (4.56) we can work out the following algebra satised by the modes T (n;m) ;T (k;r) = D (n;m) ;D (k;r) = 0 ; R n ;R y m = 0 ; T (n;m) ;R k = n k + 1 2 T (n+k;m) ; R n ;R m = (nm)R n+m ; T (n;m) ;D (k;r) =iT (n+k;m+r) ; R n ;D (k;r) =kD (n+k;r) : (4.84) Similar expressions involving the independent operator R y n are obtained by taking the Hermitian conjugate. The sub-algebra generated by T (n;m) ;R n ;R y n is the BMS 4 algebra built from su- pertranslations and superrotations. 18 Although the algebra is more complicated than the three dimensional case in (4.66), there are several qualitative similarities that result in similar represen- tations. Vacuum representation: The starting point of the vacuum representation is its invariance under ordinary conformal transformations, that gives T (n;m) j0i =D (n;m) j0i = 0 ; (n;m)2ZZ ; R n j0i =R y n j0i = 0 ; n = 0;1 : (4.85) 18 Matching with the BMS4 algebra as written in [99] involves redening some generators by adding some additional minus signs. 95 All the supertranslation and superdilation modes annihilate the vacuum due to the argument involving the ANEC in (4.59). The Hamiltonian operator that generates rigid u translations is written in terms of the supertranslation modes as H u = lim !0 1 2 Z S 2 d 2 E(z; z) =T (0;0) +T (1;1) ; (4.86) while the three rigid rotations are generated by Y 1 (z) =iz ; J 1 =R 0 +R y 0 ; Y 2 (z) = z 2 + 1 2 ; J 2 = i 2 h (R 1 R y 1 ) + (R 1 R y 1 ) i ; Y 3 (z) = i(z 2 1) 2 ; J 3 = 1 2 h (R 1 +R y 1 ) (R 1 +R y 1 ) i : (4.87) From the algebra (4.84) one can easily show these operators satisfy the appropriate SO(3) algebra [J i ;J j ] =i ijk J k . The vectors space in the vacuum representation can be spanned by acting with all the elements of the algebra on the vacuum statej0i, which results in the following states jfm i g ;f m j gi k Y i=1 R m i k Y j=1 R y m j j0i : (4.88) Using the algebra (4.84) together with the relations in (4.85) we can show these states satisfy the following properties T (n;m) jfm i g ;f m j gi =D (n;m) jfm i g ;f m j gi = 0 J 1 jfm i g ;f m j gi =M 1 jfm i g ;f m j gi ; (4.89) where the angular momentum eigenvalue M 1 in the direction of J 1 is given by M 1 k X i=1 m i + k X j=1 m j : (4.90) These relations are very similar to (4.69) for the vacuum representations in three dimensions: all the states are angular momentum eigenstates and annihilated by supertranslations and superdilations. There is a dierence coming from the fact that in the four dimensional case some of the operators contribute with a minus sign to the angular momentum, while others with a plus. We can dene states in the representation with a well dened norm in an analogous way as done in (4.70). Non-vacuum representations: Non-vacuum representations of the algebra (4.84) are obtained by starting from a statej i, which is an energy and rotation eigenstate H u j i =E u j i ; J 1 j i = 1 j i ; J 2 j i =j(j + 1)j i ; (4.91) 96 whereE u > 0, 1 2Z andj2N 0 . The HamiltonianH u is given by (4.86) whileJ 2 =J 2 1 +J 2 2 +J 2 3 is the Casimir of SO(3) built from the modes in (4.87). A straightforward computation using the algebra (4.84) shows that these operators commute, as expected. The vector space of the representation is spanned by the states obtained by acting onj i with all the elements all the algebra, which gives t Y a=1 f Y b=1 T (qa;w b ) s Y i=1 ` Y j=1 D (n i ;m j ) k Y r=1 R pr k Y r=1 R y p r j i : (4.92) Acting on these states with the angular momentum J 1 , we nd that it is an eigenstate as in (4.89) with eigenvalue given by M 1 1 t X a=1 q a s X i=1 n i k X r=1 p r + f X b=1 w b + ` X j=1 m j + k X r=1 p r ; (4.93) which generalizes the second relation in (4.89) for the vacuum representation. Similarly as in the three dimensional case, the states in (4.92) are not energy eigenstates. All in all, we see the representations of the four dimensional algebra have the same qualitative features as the three dimensional case. 4.4 Holography In previous sections, we studied several quantum aspects of the asymptotic transformations intro- duced in section 4.1. We showed the associated charges satisfy the expected algebra, and studied its representation in the Hilbert space of the CFT, as well as the action of nite transformations on the state space obtained by exponentiating the charges. In this section, we present a holographic description in the context of the AdS d+1 /CFT d correspondence. More precisely, we give a bulk geometric realization of the statesjfi,jYi andjgi dened in (4.58) and studied in section 4.3. Our starting point is the bulk description of the vacuum statej0i of the CFT, which, as usual, is identied with pure AdS d+1 . It is rst convenient to write this bulk metric using an appropriate set of coordinates such that the boundary is given by (4.9), described by the boundary coordinates (u;;~ y ). We can do this by resorting to the embedding description of AdS d+1 , starting from the surface (X 0 ) 2 (X 1 ) 2 + d+1 X i=2 (X i ) 2 =1 ; (4.94) 97 on the (d + 2)-dimensional space ds 2 =(dX 0 ) 2 (dX 1 ) 2 + d+1 X i=2 (dX i ) 2 : (4.95) The appropriate parametrization of this surface that results in the wanted boundary metric is given by X 0 = 1 +u sin( ) ; X 1 = u(2 +u) 2 sin( ) ; X 2 = +u(2 +u) 2 sin( ) ; X 3 = cot( ) j~ yj 2 1 j~ yj 2 + 1 ; X 3+A = cot( ) 2y A j~ yj 2 + 1 ; (4.96) wherey A withA = 1;:::;d2. The constraint (4.94) is automatically satised by the parametriza- tion, while the induced metric (4.95) gives the AdS d+1 space-time ds 2 AdS d+1 = 2 du 2 + 2dud +d 2 + cos 2 ( )d 2 d2 sin 2 ( ) ; (4.97) where d 2 d2 is determined by ~ y as in (4.7). The AdS d+1 boundary of the metric is obtained by taking the limit ! 0 of the coordinate 2 (0;=2] together with an appropriate rescaling, so that the boundary metric becomes lim !0 sin( ) w(x ) 2 ds 2 AdS d+1 = 2 du 2 + 2dud +d 2 d2 2 w(x ) 2 : (4.98) Depending on the value of the function w(x ) (that corresponds to choosing a conformal frame) we obtain a dierent metric at the boundary, matching with (4.9). 19 The unusual AdS d+1 coordinates (4.97) give the bulk description of the ground statej0i. The bulk horizon at = 0 corresponds to the (future) Poincar e horizon of AdS d+1 , meaning the coor- dinates (u; ;;~ y ) only cover the Poincar e patch of anti-de Sitter. See gure 4.3 for a diagram of the Poincar e patch embedded in the cylinder representing global AdS 3 . Now that we have an appropriate bulk description of the CFT ground state, we would like to extended the asymptotic (conformal) Killing vectors (4.24) for 6= 0, i.e. away from the boundary and into the bulk. For the case in which the boundary vectors generate exact conformal transformations (which amounts to taking the functions as in (4.25)) we already know how to do this, as the isometries of the bulk metric AdS d+1 give the boundary conformal transformations. 19 Note that to recover an AdS d boundary from the bulk AdS d+1 , the conformal factor is given by w(x ) 2 = sin 2 ( ^ )= 2 where ^ is dierent from the bulk coordinate in (4.97). 98 Future Poincare horizon Boundary future null innity Figure 4.3: Diagram representing the region of global AdS 3 (solid cylinder) covered by the Poincar e patch (solid blue region) in the coordinates in (4.97). While the future boundary of the bulk Poincar e patch is given by ! 0, the future null innity of the AdS boundary (the boundary of the boundary) is located at ; ! 0. The bulk Killing vectors are most easily described in terms of the embedding coordinates (4.96) as ab =X a @ @X b X b @ @X a ; a;b = 0; 1;:::; (d + 1) ; (4.99) where the index in X a is lowered using the embedding metric (4.95). Similarly as in section 4.1, let us consider the cases d = 3 and d> 3 separately. 4.4.1 Three dimensional boundary For a three dimensional boundary (d = 3), it is convenient to write y() = tan(=2) so that the AdS 4 bulk metric (4.97) takes the form ds 2 AdS 4 = 2 du 2 + 2dud +d 2 + cos 2 ( )d 2 sin 2 ( ) : (4.100) 99 The ten Killing vectors of this bulk geometry can be written in a compact way using (4.99) T (f) = cos( ) f()@ u 2 f 00 ()@ f 0 () cos( ) @ sin( )f 00 ()@ ; R (Y ) = cos 2 ( ) Y 0 ()u@ u + Y 0 ()Y 000 ()u @ + Y ()Y 00 ()u @ + + 1 2 sin(2 ) Y 0 ()Y 000 ()u @ ; S (h) = cos( )h()u 2 @ u h 0 () u(2 +u) cos( ) @ cos( ) h()2(1 +u) +h 00 ()(2 +u) 2 @ + h 00 () sin( )u(2 +u)@ ; D (g) =ug()@ u g() +ug 00 () @ ug 0 ()@ ; (4.101) where the functions that determine the vectors are given by f 0 () = a 0 cos( ) +a 1 cos() +a 2 sin() ; Y 0 () =b 0 + b 1 cos() +b 2 sin() cos( ) ; h 0 () = c 0 cos( ) +c 1 cos() +c 2 sin() ; g 0 () =d 0 : (4.102) It is straightforward to check these vectors generate exact isometries of the metric (4.100) provided the functions are xed according to (4.102). Moreover, when taking the boundary limit ! 0 for arbitrary functions, we recover the boundary vectors in (4.17) lim !0 p = p ; p =T;R;S;D : (4.103) While this suggests the bulk vectors (4.101) are the extension of the boundary vectors into the bulk, it is clear these vectors are highly non-unique. For instance, if we make the redenition D (g)! D (g) + sin( )g 0 ()@ u , the resulting vector also gives an isometry when g() = d 0 and preserves the boundary limit in (4.103). However, we favor the bulk vectors in (4.101) over these 100 other possibilities given that, when evaluated on the bulk Poincar e horizon = 0, i.e. T (f) =0 = cos( )f()@ u ; R (Y ) =0 = cos 2 ( )Y 0 ()u@ u +Y ()@ + 1 2 sin(2 )Y 0 ()@ ; S (h) =0 = cos( )h()u 2 @ u 2uh 0 () cos( ) @ 2 cos( ) h() + 2h 00 () @ 2uh 00 () sin( )@ ; D (g) =0 =g()u@ u ; (4.104) the vectorsf T (f); R (Y )g[f D (g)g satisfy a closed algebra that agrees exactly with the one satised by the boundary vectors, i.e. replacing p ! p in (4.23). 20 In other words, the boundary algebra is extended into the bulk by considering the bulk vectors (4.101) since other obvious choices typically spoil the algebra in the bulk. Using the bulk vectors p there is a natural proposal for the holographic description of the boundary states jfie iT (f) j0i ; jYie iR(Y ) j0i ; jgie iD(g) j0i : (4.105) The equivalent action of the boundary operatore iT (f) is obtained by computing the nite geomet- ric transformation associated to the bulk vector T (f), implemented by the dierential operator e T (f) . An explicit expression for the action of e T (f) is obtained by computing the integral curves associated to T (f). Our proposal is that the bulk description of the boundary states (4.105) is given by the metric obtained by acting with e p on the AdS 4 metric (4.100), so that we get g ( T )e T (g AdS ) ; g ( R )e R (g AdS ) ; g ( D )e D (g AdS ) : (4.106) The usual AdS/CFT dictionary then maps the boundary expectation values to the bulk Noether charges, see last row in table 4.6, where we summarize the holographic dictionary. To test this proposal we must compare the bulk and boundary results obtained when computing the last two rows in table 4.6 from either side of the duality. For the boundary CFT this corresponds to the results summarized in table 4.5. In the bulk we must compute two dierent quantities, the transformation of the metric itself, given by g ( p ) in (4.106) and the gravitational Noether charges. 20 Same as in the boundary analysis, including the vector S (h) results in an algebra that does not close. 101 Boundary CFT d Semi-classical gravity dual j0i g AdS p p j p ie i b Q[p] j0i g ( p )e p (g AdS ) h p j b Q[ q ]j p i Q g (p) [ q ] Table 4.6: Summary of our holographic proposal relating the boundary states (4.105) to the bulk metrics (4.106), where p =T;R;D. For the boundary charge b Q on the rst column we add a hat to remind ourselves it is an operator, written in terms of the stress tensor as in (4.30). The metric g ( p ) in the second column is given in (4.106), obtained by acting on the pure AdS d+1 metric with the (nite) transformation generated by p . Q g (p) [ q ] corresponds to the Noether charge associated to the vector q computed in the metric g ( p ). Bulk metric transformation To calculate the transformation of the metric under the action of the vectors p (as in (4.106)), we rst need to compute their associated integral curves. While for general values of the vectors in (4.101) have complicated expressions, making it very dicult to compute their integral curves explicitly, they are much simpler when evaluated at the bulk Poincar e horizon = 0 (4.104). This is analogous to the situation on the boundary, where we can only easily study the states (4.105) on the surface = 0, since the quantum charges (4.30) away from = 0 have complicated expressions. The pure AdS 4 metric evaluated at = 0 is given by ds 2 AdS 4 =0 = d 2 + cos 2 ( )d 2 sin 2 ( ) = dx 2 x 2 (1 +x 2 ) + d 2 x 2 ; (4.107) where in the second equality we have redened the bulk coordinate according tox tan( ) 0. Note that although we have not xed the time coordinate u, the induced metric is independent of u. The integral curves of the bulk vectors at = 0 can be computed analytically, so that the action of e p on the coordinates (u;;x) is given by e T (f) =0 : (u;;x) ! (u + cos( )f();;x) ; e R (Y ) =0 : (u;;x) ! p 1 +x 2 p 1 + 0 () 2 x 2 0 ()u;(); 0 ()x ! ; e D (g) =0 : (u;;x) ! (e g() u;;x) ; (4.108) 102 where the function () is implicitly dened through 21 1 = Z () d 0 Y ( 0 ) : (4.109) As a check, we take the boundary limit x! 0 and nde T (f) ande R (Y ) in (4.108) agree with the standard supertranslation and superrotation transformations in three dimensional Minkowski (see section 9.1.2 in [121]). Let us now analyze how the AdS 4 metric at = 0 (4.107) behaves under these transformations. For the special case in which we take the functionsff;Y;gg =ff 0 ;Y 0 ;g 0 g as in (4.102), the vectors p are exact Killing vectors of AdS 4 , meaning the full metric is invariant. For arbitrary functions ff;Y;gg, the supertranslated and superdilated metrics at = 0 are also invariant ds 2 T =0 g ( T )dx dx =0 = dx 2 x 2 (1 +x 2 ) + d 2 x 2 =ds 2 AdS 4 =0 ; ds 2 D =0 g ( D )dx dx =0 = dx 2 x 2 (1 +x 2 ) + d 2 x 2 =ds 2 AdS 4 =0 ; (4.110) since the AdS 4 metric at = 0 is independent of u (4.107). On the other hand, the superrotated bulk metric transforms in an interesting way ds 2 R =0 g ( R )dx dx =0 = ( 0 ()dx + 00 ()xd) 2 0 () 2 x 2 (1 + 0 () 2 x 2 ) + d 2 x 2 6=ds 2 AdS 4 =0 : (4.111) We can then apply a second transformation on the metrics g ( p ) in (4.110) and (4.111), using (4.108). While applying another transformations to the metrics g ( T ) and g ( D ) at = 0 is not dierent from considering pure AdS, the case ofg ( R ) is more interesting since the space-time at = 0 is distinct (4.111). For instance, note thatg ( R ) is not invariant under the rigid rotation ! + 0 , as the metric (4.111) depends explicitly on . This is in exact agreement with the boundary CFT result in table 4.5, where we haveR(Y 0 )jYi6= 0. The bulk results are summarized in table 4.7, where all quantities are understood to be evaluated at = 0. On the columns we write the action of the Lie derivativeL p ( ), which is obtained by expanding the nite action in (4.108) to rst order. We obtain a precise match with the boundary CFT results summarized in the rst six columns of table 4.7. In particular, the bulk computation reproduces the boundary resultj0i =jfi =jgi6=jYi. 21 See appendix A.5 for details regarding the integral curves associated to the bulk vectorR(Y ) =0 . As a concrete example, if we take Y () = cos 2 (n), the function () is dened from tan(n()) = tan(n) +n. 103 L T (f 0 ) ( ) L R (Y 0 ) ( ) L D (g 0 ) ( ) L T (f) ( ) L R (Y ) ( ) L D (g) ( ) g AdS 0 0 0 0 6= 0 0 g ( R ) 0 6= 0 0 0 6= 0 0 Table 4.7: Action of the vectors p with p =T;R;D on the bulk metric, where all the quantities in this table are evaluated at = 0. These results should be compared with the boundary CFT computations in the rst six columns of table 4.5, where we nd perfect agreement. Gravitational Noether charges We now compute the gravitational Noether charges associated to the bulk vectors p (4.101) and use the identications given in table 4.6 to compare with the boundary CFT results in the last three columns in table 4.5. To do so, we must x a particular gravitational theory, that for simplicity we take as Einstein gravity I[g ] = 1 16 Z M d 4 x p g (R + 6) + 1 8 Z @M d 3 y p hK ; (4.112) where we have included the appropriate boundary term, written in terms of the extrinsic curva- ture K. As a rst step we need to compute the gravitational charges associated to the vectors p on the pure AdS 4 space-time (4.100). An eective method that unambiguously xes its value (without the need of any vacuum subtraction) is given by the Brown-York quasi-local stress tensor [123], regulated using the counter-term method [124, 125]. The quasi-local stress tensor is dened as T quasilocal = 2 p h I onshell h ; (4.113) where I onshell is the regularized on-shell action and h is the induced metric on the surface = 0 , that we ultimately remove by taking 0 ! 0. After computing this quantity for a bulk metric g , the Noether charge Q g [ p ] associated to the vector p is computed by appropriately contracting with p and integrating (see [124] for details). For the case of the Poincar e patch of pure AdS 4 , the tensor (4.113) was computed in section 4 of [124] and shown to vanish. This means the charges on the pure AdS background are given by T quasilocal [g AdS ] = 0 =) Q g AdS [ p ] = 0 ; (4.114) in agreement with the CFT result given in the rst row and last three columns in table 4.5. 104 Q ( ) [ T (f)] Q ( ) [ R (Y )] Q ( ) [ D (g)] g AdS 0 0 0 g AdS + T g 0 0 0 g AdS + R g 0 0 1 g AdS + D g 0 1 0 Table 4.8: Gravitational Noether charges of the vectors p withp =T;R;D in (4.101) computed in a pure AdS background and the perturbed metrics in (4.117). The divergences in some of the charges arises when taking the boundary limit ! 0. Comparing with the boundary CFT results, given in the last three columns of table 4.5, we nd perfect agreement when considering quantities that do not involve the superdilation vector D (g). We add a box on the entries where there is disagreement. A less trivial calculation corresponds to computing the gravitational charges of the vectors q but on the deformed metric g ( p ) (4.106) instead. Since working with the full metric g ( p ) is very complicated, we consider its leading order behavior in p , given by g ( p ) =g AdS +L p (g AdS ) +O( p ) 2 =g AdS + p g +O( p ) 2 : (4.115) We can then calculate the Noether charge to leading order in p Q g AdS +p g [ q ] ; (4.116) where the subscript in Q indicates the background metric in which the charge is computed. To do so, it is not convenient to use the Brown-York tensor (4.113) but instead the more powerful covariant formalism. As a rst step we need to write the rst order metric variation in (4.115). Using the full expression for the vectors p in (4.101) and the AdS metric in (4.100), we nd T ds 2 =2 2 cot sin 0 ()dud + 2 2 sin ()dud 2 sin ()dd 2 0 () sin dd +::: R ds 2 =2 2 cot 2 ()du 2 2u 2 cot 2 0 ()dud 2 cot [ () +Y 0 ()]dud 2u cot [ () +Y 0 ()]dd + 2 cot Y 00 ()dd 2Y 0 ()d 2 +::: D ds 2 = 2 2 sin 2 g 00 ()du 2 u 4 sin 2 g 00 ()dud 2(cot 2 + 1 sin 2 )g 0 ()dud + 2ug 0 ()dd 2u cot 2 g 00 ()d 2 +::: (4.117) where we have dened () =f() +f 00 () and () =Y 0 () +Y 000 (). 105 Using this we can compute the charges in (4.116) for the dierent combinations ofp;q =T;R;D using the covariant formalism. Although this formalism only gives the variation of the charge between two metrics, since the pure AdS 4 metric has vanishing charge (4.114), we are actually computing (4.116) directly. An important point is that to compare with the boundary CFT results we need to evaluate the charges both at the boundary ! 0 and the null surface ! 0. This second limit is required because in previous sections we have only studied the boundary states in (4.105) on the surface = 0. The nal results for the charges are given in table 4.8, where the rst row corresponding to pure AdS comes from the Brown-York stress tensor in (4.114). We nd that all the charges vanish, in perfect agreement with the boundary CFT results, except for some cases involving the superdi- lation vector D (g), in which we get a divergence when taking the boundary limit ! 0. These divergences might be related to the fact that the superdilation boundary vector D (g) in (4.17) preserves an asymptotic condition near = 0 that is more singular than the ordinary BMS vectors, see table 4.3. Apart from this two anomalous cases, our holographic proposal for the states (4.105) at = 0 shows perfect agreement. 4.4.2 Arbitrary dimensions The above construction can be generalized to arbitrary dimensions, where the AdS d+1 metric is given by (4.97). The isometries can be once again obtained from (4.99), so the resulting vectors p that generalize (4.101) are compactly written as T (f) = cos( ) f@ u 2 d 2 (D 2 f)@ cos( ) (D A f)@ A sin( ) d 2 (D 2 f)@ ; R (Y ) = cos 2 ( ) (DY ) d 2 u@ u + cos 2 ( ) d 2 (DY ) u d 2 D 2 (DY ) @ + + Y A u d 2 D A (DY ) @ A + sin(2 ) 2(d 2) (DY ) u d 2 D 2 (DY ) @ ; S (h) = cos( )hu 2 @ u cos( ) 2(1 +u)h + (2 +u) 2 d 2 (D 2 h) @ + u(2 +u) cos( ) (D A h)@ A sin( ) u(2 +u) d 2 (D 2 h)@ ; D (g) =gu@ u g + u d 2 (D 2 g) @ u(D A g)@ A ; (4.118) 106 where D A is the covariant derivative on the unit sphere, S d2 . These vectors exactly solve the Killing equation in AdS d+1 when the functions are given by f 0 (~ y ) = a 0 cos( ) + d2 X B=1 a B y B j~ yj 2 + 1 +a d1 j~ yj 2 1 j~ yj 2 + 1 ; Y A 0 (~ y ) = b 0 y A cos( ) + d2 X B=1 p B 2y B y A AB (j~ yj 2 + 1) cos( ) + d2 X B=1 n w A B y B + ~ p B 2y B y A AB (j~ yj 2 1) o ; h 0 (~ y ) = c 0 cos( ) + d2 X B=1 c B y B j~ yj 2 + 1 +c d1 j~ yj 2 1 j~ yj 2 + 1 ; g 0 (~ y ) =d 0 ; (4.119) where w A B =w B A is an antisymmetric matrix. As we take the limit ! 0, both the vectors (4.118) and functions (4.119) reduce to the boundary quantities in (4.24) and (4.25) that generate conformal transformations on the boundary. Moreover, when evaluating the vectors at the bulk Poincar e horizon = 0 we nd T (f) =0 = cos( )f(~ y )@ u ; R (Y ) =0 = cos 2 ( ) (DY ) d 2 u@ u +Y A @ A + sin(2 ) 2(d 2) (DY )@ ; S (h) =0 = cos( )h(~ y )u 2 @ u 2u(D A h) cos( ) @ A 2 cos( ) h(~ y ) + 2 d 2 (D 2 h) @ 2u sin( ) d 2 (D 2 h)@ ; D (g) =0 =g(~ y )u@ u : (4.120) Computing the algebra associated to these vectors (without the vector S (h)) we nd it is exactly the same as the boundary algebra, i.e. it becomes (4.29) after replacing!. Following the same criteria as in the three dimensional case, the boundary vectors p are extended inside the bulk by p in (4.119). The holographic description of the boundary statesfjfi;jYi;jgig parallels the one described for the AdS 4 case, summarized in table 4.6. While the computation of the transformed bulk metric g ( p ) in (4.106) and the Noether charges is more involved, we expect analogous results as those given in tables 4.7 and 4.8. 107 4.5 BMS and black holes Before concluding, let us make some comments about the possible applications that these results could have to investigate black hole physics. In fact, the main motivation for studying the action induced by the asymptotic (conformal) Killing vectors on the Hilbert space of a CFT is to get further insight on quantum aspects of black holes. The study of BMS symmetries in relation to black holes goes back to the proposal made in [110], and it is natural to ask whether the particular realization of the innite-dimensional symmetry we studied here could have something to do with it. The connection arises for (near-)extremal black holes, that has a near horizon limit that corresponds to AdS 2 S d2 . For concreteness, let us focus on the case of asymptotically at electrically charged black holes in four dimensional Einstein-Maxwell theory I[g ;A ] = 1 16G Z d 4 x p gR 1 4 Z d 4 x p gF F ; whose metric is given by the Reissner-Nordst om geometry ds 2 =f(r)dt 2 + dr 2 f(r) +r 2 d 2 2 ; f(r) = (rr + )(rr ) r 2 ; (4.121) where r are the inner and outer horizons, which can be written in terms of the mass and charge of the black hole. The extremal and near-extremal black holes correspond to r + =r andr + 'r respectively. If we study the near horizon limit r r + , in both cases we nd it is given by AdS 2 S d2 . However, since the bifurcation surface in the extremal and near-extremal case ends up in a dierent place of the AdS 2 factor, we must analyze each case separately. A detailed analysis of the near horizon limit of this black hole can be found in [2]. Let us start by considering the extremal black hole r + = r that has a maximally extended Penrose diagram shown in the left diagram of gure 4.4. The diamond region corresponds to the exterior of the black hole where the initial value problem can be dened. We sketch a Cauchy surface t given by a constant t surface on the black hole space-time (4.121) with r + =r . There are four boundaries, the green lines on the left corresponding to the future and past null innity I , and the blue dashed lines to the future and past black hole horizons H . Taking the near horizon limit of the extremal solution (4.121) we get an AdS 2 S d2 metric, where the black hole 108 AdS Boundary AdS Boundary Near Horizon Limit Figure 4.4: On the left we see the maximally extended Penrose diagram of the extremal Reissner- Nordstrom black hole. Taking the near horizon limit we obtain the AdS 2 S d2 space-time on the right, see [2]. The gray shaded region corresponds to the section of the black hole that is well approximated by AdS 2 S d2 . horizonsH correspond to the surfaces = in the global coordinates given in (4.11). As shown in the right diagram of gure 4.4, the near horizon limit is equivalent to slicing the Penrose diagram of the black hole by inserting the two AdS 2 boundaries, one on the singularity would be and the other along the surface whereI and H meet. Let us now consider an arbitrary CFT dened on the exterior region of the extremally charged black hole. For any Cauchy surface t we have an associated Hilbert spaceH t . We can move through dierent foliations, i.e. dierent values of t, by acting on the t-translation operator as e t@t : t 1 ! t 2 , and so e itQ[@t] : H t 1 !H t 2 , where Q[@ t ] is the conserved charge (4.30) written in terms of the stress tensor operator of the CFT. Since t-translation is an isometry of the black hole metric (4.121), the vacuum of the theory, which we denotej i, is invariant under such a transformation, namely e iQ[@t] j i =e iE 0 j ij i, where E 0 is the energy of the ground state. There are two Cauchy surfaces that are of particular interest to us. These are 22 initial t!1 = I [H and nal t!+1 =I + [H + , cf. [110]. On these Cauchy surfaces we expect the vacuum statej i to coincide with the vacuum of the CFT dened in Minkowski and AdS 2 S d2 , so that we can act with the asymptotic chargesR(Y ) to generate the zero energy eigenstatesjYi 22 The factorization of the space on the Cauchy surface has been recently discussed in [126, 127]. 109 Conformal Transformation Conformal Transformation CRT Transformation CRT Transformation Figure 4.5: Diagram showing the transformations that map between the four boundaries of the extremal black hole (4.121): I and H . The mapping of the charges under the conformal and CRT transformations where explored on the rst two subsections of section 4.2. with respect to the Hamiltonians that generates the time evolution in the u and v coordinates. Interestingly, the states on the surfacesI andH are not independent but related by confor- mal symmetry (mapping the Minkowski surfaceI to the Poincar e horizons H in AdS 2 S d2 ) and CRT symmetry (mapping between the asymptotic regionsI themselves). The transforma- tions relating the dierent surfaces on the exterior of the black hole are summarized in gure 4.5. Using these transformations we can relate the zero energy statesjYi on the dierent surfaces (see (4.55)). Note that when applying the CRT transformation the function Y A (~ y ) is not invariant but transforms in the way specied in (4.54). One might still worry about the transformations in gure 4.5 not being well behaved as they only involve the near horizon and asymptotic regions, i.e. they are not true transformations of the whole black hole space-time. However, for this particular type of black hole, we can actually go even further and apply a conformal transformation on the full metric (4.121) that maps the asymptotic regionsI to the horizons H . This is obtained by considering [128] r = r r=r + 1 ; (4.122) which yields ds 2 = 1 ( r=r + 1) 2 f( r)dt 2 + d r 2 f( r) + r 2 d 2 2 ; (4.123) where we have to remember that we are considering the extremal case r + =r ; see also [129]. Ap- plying a Weyl transformation which removes the conformal factor, we recover the four dimensional 110 Near Horizon Limit AdS Boundary AdS Boundary Figure 4.6: On the left we see the maximally extended Penrose diagram of the near-extremal Reissner-Nordstrom black hole. Notice that in this case the future (past) horizon H + (H ) is actually the past (future) horizon for the asymptotic region on the right. Taking the near horizon limit we obtain the AdS 2 S d2 space-time on the right, see [2]. The gray shaded region corresponds to the section of the black hole that is well approximated by AdS 2 S d2 . black hole, with the dierence that the horizon r = r + has been mapped to the asymptotic re- gion r! +1 and viceversa. This transformation is very special to the extremal four dimensional Reissner-Nordstr om solution, and it does not generalize to higher-dimensional or non-extremal black holes, at least not in an obvious manner. Therefore, at least in the case d = 4, it is natural to ask whether this conformal symmetry together with the symmetries studied in this chapter can be used to construct the zero energy states on the full black hole geometry. A similar discussion applies to near-extremal black holes withr + 'r , that are more interesting as they have nite temperatureT + . The Penrose diagram of the black hole solution is given on the left diagram in gure 4.6, where we see that in this case there are two asymptotically at regions that are causally disconnected from each other. The exterior black hole space-time described by the metric (4.121) is shaded on the left side. The near horizon limit r r + in this case also yields an AdS 2 S 2 metric but only after we apply a change of coordinates (see (2.6) in [2]). As we see on the right diagram in gure 4.6, the dieomorphism changes the location of the horizons H , as they now intersect in the interior of AdS 2 , at (;) = (0;=2) in the global coordinates (4.11). The exterior of the black hole covered by the AdS 2 does not correspond to the Poincar e patch as in the extremal case, but to the smaller region <=2. 111 The position for the horizons in AdS 2 S 2 is obtained by taking 0 = =2 when dening the coordinates (u;) and (v;%) in (4.10) and (4.46) respectively. The horizons H appearing in the right diagram of gure 4.6 correspond to = 0 and % = 0 in each of these coordinates. Note that these coordinates cover a larger region than the one corresponding to the exterior of the black hole <=2. Similarly to the extremal case, we can dene the BMS transformations on H andI and a family of zero energy states by acting withR(Y ) on the vacuum. The states dened in each of these null surfaces are related between themselves in a similar way as described for the extremal case in gure 4.5. 112 Part II Matrix models and quantum gravity 113 Chapter 5 Introduction In this second part of the thesis we study two dimensional theories of quantum gravity, where several of the obstacles that arise when trying to quantize gravity in higher dimensions are absent. Recent progress has shown it is possible to construct certain simple models whose observables can be computed unambiguously. Remarkably, the perturbative expansion of those observables is exactly reproduced by certain ensembles of random matrices, to all orders in perturbation theory. Given that ensembles of random matrices have been studied for a long time and are very well understood, a fully non-perturbative denition of the quantum gravity theory can be given in terms of matrices. In this rst chapter we provide a general introduction which motivates the connection between matrix models and quantum gravity. 5.1 Topological expansion Perhaps the simplest observable one can try to compute in a theory of quantum gravity is the partition function, formally dened from the following path integral Z = Z Dg De I E [g;] ; (5.1) where represent all the matter elds present in the theory and I E [g ; ] the Euclidean action. The path integral is supposed to integrate over all smooth congurations (g ; ) consistent with the boundary conditions, which must be specied when dening Z. One of the advantages of working with a two dimensional theory is that the partition function 114 admits what is usually called a \topological expansion". This builds on the observation that any two dimensional metric automatically satises Einstein's equations R 1 2 Rg = 0 ; (5.2) whereR is the Ricci tensor andR its trace. As a result, adding the Einstein-Hilbert action to I E [g ; ] does not modify the equations of motion. Due to the Gauss-Bonnet theorem this additional term in the action is nothing more than Euler's characteristic (M) = 1 4 Z M d 2 x p gR = 2(1g) ; (5.3) where g2N 0 is the genus of a closed and orientable manifoldM. Using this, we can write the following expansion for the partition function (5.1) Z = 1 X g=0 (e S 0 ) 2(g1) Z g ; (5.4) whereS 0 2R + is the coupling constant that multiplies the Einstein-Hilbert term in the Euclidean action. Each term in this expansion is determined byZ g , that is the same partition function in (5.1) with the dierence that instead of integrating over all possible metrics, we only include manifolds of xed genusg. Although this might not seem a huge simplication, in the following chapters we will exhibit several examples in which all terms in the topological expansion can be computed exactly. It is important to point out that from the gravity perspective, successive terms in the expansion Z g and Z g+1 are related to one another non-perturbatively, i.e. there is no small perturbation one can apply to the metric on a sphere S 2 , of genus g = 0, that can transform it into a torus S 1 S 1 of genus g = 1. While the topological expansion in (5.4) is an extremely powerful and convenient way of orga- nizing the calculation, it has certain important drawbacks. To start, note that once the action and boundary conditions are xed the full partition function Z is nothing more than a (typically very complicated) function of the parameter S 0 . As such, the topological expansion is a perturbative series ine S 0 that will contain all the information aboutZ only if the series has an innite radius of convergence. General arguments suggest it is not a convergent but an asymptotic series [130] (see also [131, 132]). This means there are (possibly important) non-perturbative contributions to Z that are not captured by the expansion in (5.4). From the gravity perspective, these correspond to 115 doubly non-perturbative eects, of orderO(e e S 0 ). In the remaining of this thesis we shall explore a dierent approach for computingZ using an ensemble of random matrices that captures not only the topological expansion in (5.4) but also the full dependence onS 0 of the partition function (5.1). 5.2 Tesselating surfaces and random matrices An alternative approach for computing the partition function (5.1) (see [133{137] for some early references) is to consider a discrete version of the path integral over metrics. Our discussion here follows [138] and is not supposed to be rigorous but understood as empirical evidence for the relation between two dimensional quantum gravity and random matrices. In the next chapters we shall see how these ideas become rigorous and lead to concrete calculations and results. One way of discretizing a manifoldM is by constructing a tesselationT , dened to be a covering ofM by any collection of polygons. Each polygonP is built from certain vertices and edges which t entirely in some open set U M where we can take some coordinates :U !R 2 , so that after the mapping P is an ordinary at polygon inR 2 . Given two polygons P i and P j their only possible intersection is along a common edge or vertex. It can be shown that any smooth manifold admits a tesselationT that is build entirely from triangles (see [139] for details). We also need the tesselation to capture the orientation of the manifoldM. To do this, we pick a particular polygon P and assign an arbitrary orientation. The orientation for all the other polygons in the tesselation is assigned in such a way that adjacent polygons have opposite orientation along their common edge, see the center diagram in gure 5.1. Starting from a simple tesselationT with a few number of polygons P ,T can be modied by including additional vertices and edges so that we obtain a rened tesselationT 0 . In this way one can construct a rened tesselation that better approximates the dierentiable manifoldM. In gure 5.1 we show three dierent tesselations built only with triangles of a unit sphere S 2 , where we observe how the renement results in a better approximation to the smooth manifold we are ultimately interested in. The intuitive idea is that we can use this construction to replace the path integral over metrics in Z (5.1) by a summation over all possible tesselationsT Z Dg ! X T (5.5) 116 Figure 5.1: Three dierent tesselations of a unit sphere S 2 , built entirely in terms of triangles. As the tesselation is rened we observe how it is able to more accurately approximate the smooth surface of the dierentiable manifold. In the center diagram we indicate the orientation of some of the triangles. where the summation on the right is usually restricted to a particular class of polygons P , e.g. only triangles. Although this substitution is neither rigorous nor guaranteed to work, the hope is that it provides a provides a computable quantity that approaches the true value of the path integral in the limit in which the sum is performed over highly rened tesselations. To make sense of this procedure, we also need to write a discrete version of the terms that appear in the action I E [g ; ] in (5.1). For a couple of simple examples they can be written as 1 4 Z M d 2 x p gR =(M) =V +FE ; Z M d 2 x p g = Area(M) = F X =1 a ; (5.6) where V , F and E are the number of vertices, faces and edges of the tessellationT , while a the area of each polygon. It is not hard to show that the Euler characteristic (M) as written above is independent of the particular tesselation, i.e. it is invariant under the addition of a vertex or edge. 1 Moreover, one can explicitly use this formula on a sphere, torus and double-torus to show (M) = 2(1g), as expected. Note that as the tesselation is further rened and the number of faces F increases, the area of the individual polygonsa decreases (see gure 5.1). By taking an appropriate double scaling limit F 1 and a 1 we should recover the dierentiable manifoldM. We expect this result to be universal, in the sense that it should be independent of the details of the discrete tesselation, e.g. 1 Adding n additional edges E = n leaves (M) in (5.6) invariant given that it generates F = n additional faces while keeping the number of vertices xed V = 0. On the other hand, if we add a vertex one must at least include E =m 2 in order to avoid having a disconnected vertex or edge, which results in F =m 1 meaning (M) is unchanged. 117 it should be the same whether we use triangles or squares to cover the manifold. Let us now discuss an ensemble of Hermitian random matrices M, which turn out being useful for computing the discrete sum over tesselations [140, 141]. The partition function for the model of interest is dened as Z(;N) = 1 Z 0 Z dMe N Tr [ 1 2 M 2 M 4 ] = 1 X q=0 q q! 1 Z 0 Z dMe N 2 TrM 2 N TrM 4 q ; (5.7) where N is the size of the matrix and a coupling constant. 2 The integration measure dM is invariant under change of basis of M and is dened below in (6.1). The normalization constant Z 0 is the integral in the numerator with = 0. To evaluate each of the terms in the series it is convenient to dene the following expectation value with respect to the Gaussian measure hOi = 1 Z 0 Z dMe N 2 TrM 2 O ; (5.8) whereO is an arbitrary function ofM. Instead of computingZ(;N) we are going to be interested in the free energyF(;N) dened as F(;N) = lnZ(;N) = 1 X q=0 q q! h N TrM 4 q i c ; (5.9) where the subscriptc indicates it is the connected expectation value dened from (5.8). Each term in this expansion can be computed using Wick's theorem. Everything is determined by the two point functionhM ij M kl i, that is obtained from the inverse of the quadratic term in the exponent of (5.8) hM ij M kl i = 1 N il jk : (5.10) Using this, we can compute the series (5.7) to whatever order we desire. For instance, the rst non-trivial term gives hN TrM 4 i =N N X i;j;k;l=1 hM ij M jk M kl M li i =N 12 N X i;j;k;l=1 ik jj ki ll + ii jl jl kk + il jk ji kl =N 2 +N 2 + 1 : (5.11) 2 In the second equality we have exchanged the integral with the series, which is not always allowed, specially for positive when the integral is not convergent. The partition functionZ(;N) can be directly dened from the second equality as what is sometimes called a \formal matrix model", see [142]. 118 Figure 5.2: The three ribbon diagrams that contribute toF(;N) in (5.9) to linear order in . The rst two diagrams can be drawn on the plane (or a sphere) and contain two additional closed loops, indicated in red and green. The third diagram can be drawn on a torus and does not contain any additional loop. Each of these give the three terms in (5.11). For our purposes, it is actually more interesting to recover this result using Feynman diagrams, which are constructed from the following two building blocks M ij = TrM 4 = (5.12) These are ribbons diagrams, with two lines instead of one, since we must keep track of the two indices of the matrix M. Moreover, since the matrix is not symmetric, we add an arrow to each line that dierentiates between the rst and second index. Using this, it is quite easy to recover the answer in (5.11), which involves taking all possible contractions of the four legs in the vertex (5.12), using that the propagator is given by (5.10). The three diagrams that contribute are shown in gure 5.2. The rst two diagrams give a factor ofN 2 that comes from the sum over the additional closed loops indicated in red and green. In this way, we recover the answer previously computed in (5.11). The connection with our previous discussion comes from the observation that these three di- agrams represent all the diagrams with one vertex that can be drawn on a surface of arbitrary genus. For a sphere g = 0 (or plane) there are two diagrams that contribute with N 2 , while for the torus g = 1 there is a single one that adds the term N 0 . For higher genus surfaces no diagram can be drawn with only one vertex. Essentially, the expectation value in (5.11) counts the number of possible tessellations of a surface using a polygon with four edges, i.e. a square. If we had started with the partition function (5.7) but with M p in the exponent, we would have obtained 119 tessellations built from a polygon with p edges instead. To compute the tessellations built from a q number of squares we need to compute the q-th term in (5.9). In terms of Feynman diagrams, these are obtained from all possible contractions of h N TrM 4 q i c = (5.13) which gives rise to (2q + 1)!! diagrams. Each of these diagrams contributes with a factor of N that depends on the number of closed loops Diagram =N q 1 N q1 1 N q+1 N #loops : (5.14) The rst factor comes from the q vertices, while the second one from the propagators needed to connect the vertices already connected in (5.13). The third factor arises from the propagators needed to connect the remaining 2q + 2 external legs. The number of loops depends on the surface on which each diagram can be drawn in, for instance Spherical diagrams =N q 1 N q1 1 N q+1 N q+2 =N 2 ; Torus diagrams =N q 1 N q1 1 N q+1 N q = 1 ; Double torus diagrams =N q 1 N q1 1 N q+1 N q2 = 1 N 2 : (5.15) From this we recognize that for each diagram we have N 2(1g) where g is the genus of the surface where the diagram can be drawn. In gure 5.3 we sketch three diagrams that contribute when q = 3 and can be drawn on a surface with genus g = 0; 1; 2. Altogether, for an arbitrary q we have the following expansion h N TrM 4 q i = b q+1 2 c X g=0 N 2(1g) C q;g ; q 1 ; (5.16) where C q;g is the number of diagrams with q vertices that can be drawn on a surface of genus g. The free energy of the matrix modelF(;N) in (5.9) can be written using this expansion as F(;N) = 1 + 1 X q=1 b q+1 2 c X g=0 q q! N 2(1g) C q;g = 1 + 1 X g=0 N 2(1g) 1 X q=j2g1j q q! C q;g ; (5.17) 120 Figure 5.3: Three ribbon diagrams that contribute toh N TrM 4 3 i, where the number of closed loops is indicated with dierent colors. Each of these diagrams contribute with N 2 , N 0 and N 2 respectively, and can be drawn on a sphere, torus and double torus. Below each diagram we indicate the contraction of the external legs in (5.13) required in each case. where in the second equality we have exchanged the summations. It is now that we can compare with the topological expansion of the gravity partition function in (5.4) and note some resemblance, where 1=N is playing the role ofe S 0 . The matching is not precise, given that for any given genus g, the factor between square brackets in (5.17) gets contributions from diagrams with a small number of polygons, since q can be as small asj2g 1j. From the gravity perspective this is not what we need, given that small q corresponds to tessellations that are not rened. For instance, for the case of the sphere the low q terms in (5.17) correspond to tessellations like the one in the left diagram of gure 5.1, where instead we would like to only have tessellations that resemble the diagram on the right. We therefore need the diagrams with a very large number of polygons to dominate over terms with just a few. In other words, writing the expansion forF(;N) as F(;N) = 1 + 1 X g=0 N 2(1g) F g () ; (5.18) we want F g () to be dominated by diagrams with a large number of vertices. To do so, let us replace in the denition of the matrix partition function (5.7) by ! ( c ) where c is a critical value which ensures the following behavior for F g () F g () =v g ( c ) n m +::: ; (5.19) where v g is some constant and n m = 2 Z. Subleading terms in this expansion are regular, i.e. an integer power of ( c ), or singular but subleading. Calculating a series expansion for F g () centered around c , higher order terms will diverge with a stronger divergence. This is precisely 121 the behavior we are looking for, as from (5.17) it implies the diagrams with a larger number of vertices (the more rened tessellations) are the ones that dominate for each genus g. Altogether this suggests the partition function in quantum gravity (5.1) can be computed from the matrix model in the following way Z quantum gravity ! F(;N) ; c ; N 1 : (5.20) This double limit in the matrix model is called \double scaling limit" and was rst developed in the nineties [143{145]. There are several questions that are left unanswered by this construction. Can we actually nd a critical value c which displays the critical behavior required in (5.19)? There are an innite number of two-dimensional gravity theories that can be dened from the Euclidean action in (5.1), which can also include some matter elds. Is there always a matrix model that can be constructed so that its partition function is reproduced in this way? If so, what is the correct denition of the probability distribution (5.7) of the matrix model for each case? Finally, can we use the matrix model to provide a non-perturbative denition of the quantum gravity theory, beyond the topological expansion (5.4)? In this part of this thesis we attempt to partially answer some of this questions. We start in the following chapter by carefully and precisely introducing matrix models and their double scaling limit. In the chapter 7 we introduce a class of dilaton gravity theories called Jackiw- Teitelboim gravity and supergravity which have been shown in [3, 146] to be well described by double scaled matrix model. In particular, we show how the random matrices can be used to provide a non-perturbative denition of the quantum gravity theory that allows for concrete and explicit calculations. 122 Chapter 6 Random matrix models In this chapter we introduce two dierent methods for computing observables in an ensemble of random matrices: the loop equations and the method of orthogonal polynomials. We discuss the double scaling limit of these models, that is the regime in which they become useful for describing two dimensional quantum gravity. Consider an ensemble of square Hermitian matricesM of dimensionN, weighted by a probability measure determined by a potential V (M) according todMe N TrV (M) . The measuredM includes all the real and independents components of the matrices dM = N Y i=1 dM ii Y i<j dRe(M ij )dIm(M ij ) ; (6.1) so that the expectation value of any arbitrary matrix operatorO is dened as hOi 1 Z Z dMOe N TrV (M) ; where Z = Z dMe N TrV (M) : (6.2) Note the exponential factor TrV (M), which eectively denes a particular matrix model, is in- dependent of the particular basis we choose to write the matrix M. This is also the case for the measure dM in (6.1), which satises dM =d(UMU y ) for any unitary matrix U. 1 For this reason, it is convenient to parametrize the independent components of M in terms of (;U) dened as M = UU y , where = diag( 1 ;:::; N ) with i 2R. The unitary matrix U can be written as U = e iL , with L hermitian. In order to avoid over counting we should take U in the coset group 1 It is instructive to work out a simple case explicitly, for instance by using the any two dimensional unitary matrix can be written as U =e i e i 1 cos() e i 2 sin() e i 2 sin() e i 1 cos() . 123 U(N)=U(1) N , corresponding to L Hermitian with vanishing diagonal elements. As a result, the change of variables is given by M ii ; Re(M ij ); Im(M ij ) ! i ; Re(L ij ); Im(L ij ) ; j 0 and 2 p 4 =3 < 2 < 0. The dashed gray line corresponds to the corresponding potential V (z) = 2 z 2 + 4 z 4 =3. On the right we plot the contours in the complex plane z 0 used to obtain the recursion relations from the loop equations. For z 0 2 (a ;a + ) we observe the branch cut in the spectral curve y(z 0 ). case we have(z) = (za )(za + ) wherea 2R determine the location of the branch cut. This can be seen more explicitly by using (6.11) to compute the eigenvalue spectral density from the discontinuity of W 0 (z) in the complex plane, which gives 0 () = 1 2 h() p () 1 ()<0 ; (6.24) where 1 ()<0 is the indicator function. For any given potential we can work out a and h(z) by requiringW 0 (z) in (6.23) has the appropriate large z behavior, i.e. W 0 (z) = 1=z +O(1=z 2 ). As an example, for the potential V (z) = 2 z 2 + 4 z 4 =3 with 4 0 we nd h(z) = 2 3 q 2 2 + 4 4 + 2 2 + 2 4 z 2 ; a 2 + =a 2 = p 2 2 + 4 4 2 4 : (6.25) On the left diagram of gure 6.1 we plot the spectral density and potential. Note the spectral density is positive denite for 4 > 0 and 2 >2 p 4 =3. For smaller values of 2 there is a phase transition to a double-cut spectral density, i.e. supported on two disjoint intervals instead of a single one. Going back to the denition of the spectral curve y(z) in (6.21), we see it can be written as y 2 (z) = h(z) 2 (z)=4. This denes a Riemann surface with two sheets, corresponding to the two possible signs of the square root. If we denote ^ z as the same point asz but in the second sheet, we have the following relations h(^ z) =h(z) ; p (^ z) = p (z) ; y(^ z) =y(z) : (6.26) As we shall see below, the whole perturbative expansion of W (I) can be xed in terms of the spectral curve y(z). For this reason, a perturbative denition of the matrix model can be given 128 directly in terms of y(z) instead of the potential V (z). From (6.11) we note the spectral curve is very much related to the leading spectral density 0 () 0 () = i y(i) : (6.27) 6.1.2 Recursion relation Let us now show how the loop equations in (6.20) can be used to derive a general recursion relation that determines all observables in the matrix model perturbatively. A universal observable We start by taking g = 0 and I =fz 1 g in the loop equations (6.20) and solving for W 0 (z;z 1 ) W 0 (z;z 1 ) p (z) = p (z) 2y(z) 1 + (zz 1 )@ z 1 (zz 1 ) 2 W 0 (z 1 )P 0 (z;z 1 ) V 0 (z) 2(zz 1 ) 2 p (z) 2(zz 1 ) 2 ; (6.28) where we have used the denition of the spectral curve y(z) in (6.21). While this expression might seem cumbersome, it turns out being very useful. Note that the terms between square brackets on the right are meromorphic in z, i.e. they are analytic up a pole for z! z 1 . Moreover, the combination p (z)=2y(z) is also single valued, with possible poles at the roots of h(z). The last term on the right-hand side is quite dierent, given that it not only contains a pole at z! z 1 but also branch cut along z2 (a ;a + ). We can take advantage of the simple analytic structure of W 0 (z;z 1 ) p (z) by appling Cauchy's integral formula in the following way W 0 (z;z 1 ) = 1 2i p (z) I z dz 0 z 0 z W 0 (z 0 ;z 1 ) p (z 0 ) = 1 2i p (z) I C dz 0 z 0 z W 0 (z 0 ;z 1 ) p (z 0 ); (6.29) where in the second step we have deformed the contour in the complex plane toC (right diagram in gure 6.1). The contour deformation does not pick up any additional contributions sinceW 0 (z 0 ;z 1 ) is by construction analytic in the complex plane except for z 0 2 (a ;a + ). Using (6.28) we get a huge simplication, given that the rst term on the right vanishes since it is the integral of an analytic function insideC. 4 We are thus left with a straightforward integral W 0 (z;z 1 ) = 1 2i p (z) I C dz 0 z 0 z p (z 0 ) 2(z 0 z 1 ) 2 = 1 2i p (z) Z a + a dx 0 x 0 z p (x 0 ) (x 0 z 1 ) 2 ; (6.30) 4 We are assuming h(z) does not vanish for z2 [a;a+], which is equivalent to saying the system is not critical. For the simple example we worked out in (6.25) it means 26= 2 p 4=3 so that h(0)6= 0. 129 which can be easily solved to give W 0 (z;z 1 ) = 1 2(zz 1 ) 2 " a a + +zz 1 (a +a + )(z +z 1 )=2 p (z) p (z 1 ) 1 # : (6.31) The remarkable feature of this result is that it is independent of the details of the potential V (z) and spectral curve y(z). It is universal, meaning it only depends on the endpoints a of 0 (). Before moving on, let us note the value of W 0 (z;z 1 ) on the two sheets of the Riemann surface are related according to 2W 0 (^ z;z 1 ) + 1 (^ zz 1 ) 2 = 2W 0 (z;z 1 ) + 1 (zz 1 ) 2 ; (6.32) where we have used (6.26). Recursion relation This same approach can be applied to computeW g (z;I). First, we use the loop equations in (6.20) to solve for W g (z;I) W g (z;I) p (z) = p (z) 2y(z) 2 4 jIj X k=1 1 + (zz k )@ z k (zz k ) 2 W g (I)P g (z;I) 3 5 p (z) 2y(z) F g (z;I) ; (6.33) where we have dened F 0 (z;z 1 ;z 2 ) = W 0 (z;z 1 ) (zz 2 ) 2 + W 0 (z;z 2 ) (zz 1 ) 2 + 2W 0 (z;z 1 )W 0 (z;z 2 ) ; (6.34) for g = 0 and I =fz 1 ;z 2 g and F g (z;I) =W g1 (z;z;I) + jIj X k=1 2W 0 (z;z k ) + 1 (zz k ) 2 W g (z;Inz k ) + 0 X h;J W h (z;J)W gh (z;InJ); (6.35) otherwise. The sum in the third term is over allh = 0;:::;g andJI that do not contain a factor of W 0 (z) or W 0 (z;z k ). The case with g = 0 and I =fz 1 ;z 2 g in (6.34) has to be treated separately as this combination is special when it comes to extracting the terms that contain W 0 (z;z k ) in the summation in the second line of (6.20). The next step is to apply Cauchy's integral formula in exactly the same way as before (6.29) but replacingW 0 (z 0 ;z 1 )!W g (z 0 ;I). Using (6.33) and noting the rst term on the right is analytic in the interior ofC, we nd W g (z;I) = 1 2i p (z) I C dz 0 z 0 z p (z 0 ) 2y(z 0 ) F g (z 0 ;I) : (6.36) 130 Since F g (z 0 ;I) only depends on W g 0(I 0 ) with either g 0 < g or I 0 I, this gives an expression for W g (z;I) in terms of the lower order expansion coecients. The integral in the complex z 0 plane can have contributions from poles at z 0 =a or a branch cut along z 0 2 (a ;a + ). Let us rst look at the branch cut, by noting (6.26) implies the factor p (z 0 )=2y(z 0 ) takes the same value along both sides of the cut. For the caseg = 0 andI =fz 1 ;z 2 g we can use (6.32) to show F 0 (z;z 1 ;z 2 ) in (6.34) is also single valued, meaning W 0 (z;z 1 ;z 2 ) only gets contributions from the branch points at z 0 = a . To show the same applies for the higher order terms, let use induction to show the following property W g (^ z 1 ;z 2 ;:::;z n ) =W g (z 1 ;z 2 ;:::;z n ) ; (6.37) is satised in all cases but W 0 (z 1 ) and W 0 (z 1 ;z 2 ). 5 This property is certainly satised by W 0 (z;z 1 ;z 2 ) since the residues at z 0 = a are analytic in z and the minus sign in (6.37) comes from the prefactor 1= p (z) in (6.38). Assuming (6.37) is true for some xed value of g andn, one can showW g (z;I) as computed from (6.38) satises the same property. To start we use (6.37) and (6.32) to prove all the terms of F g (z;I) in (6.35) are single valued. Therefore, the integral (6.38) only gets contributions from the poles at z 0 = a , which are analytic in z. Same as before, the prefactor 1= p (z) gives the minus sign in (6.37). This analysis greatly simplies the computation of W g (z;I), as it does not really involves an integral but just some residues at z 0 =a , which can be written as W g (z;I) = 1 p (z) X i= Res " p (z 0 ) 2y(z 0 ) F g (z 0 ;I) (z 0 z) ;z 0 =a i # (6.38) This is the nal form of recursion relation for the expansion coecients W g (z;I), determined just from the knowledge of the spectral curve y(z) and the location of the branch points a . Using p (z 0 )=2y(z 0 ) = 1=h(z 0 ), we can easily work out the rst few cases and nd W 0 (z 1 ;z 2 ;z 3 ) =(a + a ) h 1 (z 1 a )(z 2 a )(z 3 a )h(a ) 1 (z 1 a + )(z 2 a + )(z 3 a + )h(a + ) i 8 p (z 1 ) p (z 2 ) p (z 3 ) ; W 1 (z) p (z) = (2z +a 3a + )h(a + ) + (a + a )(za + )h 0 (a + ) 16(a + a )(za + ) 2 h(a + ) 2 (2z +a + 3a )h(a ) (a + a )(za )h 0 (a ) 16(a + a )(za ) 2 h(a ) 2 : (6.39) 5 These two special cases is the reason we had to write F0(z;z1;z2) and Fg(z;I) in (6.34) and (6.35) separately. 131 Comparing with the universal result for W 0 (z 1 ;z 2 ) in (6.31), these expressions do not only depend on a but on h(z) and its derivatives at the endpoints. 6.1.3 Double scaling limit The random matrix models that arise as descriptions of two dimensional quantum gravity are subject to a particular limit called \double scaling". From the perspective of the loop equations this corresponds to models for which a + or/and a go all the way to innity. One of the consequences of this is that the spectral density 0 () is no longer normalizable. Introducing a small parameter ~ and scaling the spectral density according to 0 ()! 0 ()=~, the 1=N expansion is replaced by an~ expansion 6 hW (I)i c = 1 X g=0 ~ 2(g1)+jIj W g (I) : (6.40) Apart from this, all the technology works in the same way as for an ordinary matrix model. The computation of the residue (6.38) at innity greatly simplies. Taking a + ! +1 it is given by 1 p (z) Res " p (z 0 ) 2y(z 0 ) F g (z 0 ;I) (z 0 z) ;z 0 ! +1 # = 1 p z Res 1 2 p z 0 y(1=z 0 ) F g (1=z 0 ;I) z 0 (1zz 0 ) ;z 0 = 0 ; (6.41) where for simplicity we have also xeda = 0. The function p z 0 y(1=z 0 ) in the denominator, which is analytic in 1=z 0 , gives a contribution that is regular atz 0 = 0. Using thatW (z) = 1=z +O(1=z 2 ) we also have F g (1=z;I) = 1=z 2 +O(1=z 3 ), so that the residue at a + ! +1 always vanishes, independently of the spectral curve y(z). This simplication is even more dramatic when the double scaled model has both a and a + at innity, as in that case we get lim a !1 W g (z;I) = 0 ; (6.42) apart from the two special casesW 0 (z) andW 0 (z 1 ;z 2 ) in (6.31). This observation plays an important role when using this kind of double scaled matrix model to describe a particular supergravity theory in the following chapter. Before nishing this section, it is instructive to work the rst few terms in the expansion of a simple double scaled model, with a spectral density given by y(z) = p z. The leading spectral 6 By ~ here we do not mean the actual Planck constant, but a small parameter that controls the expansion of observables. The reason for calling this~ will become clear in the following section. 132 density of this model (sometimes called the \Airy model") can be easily computed and is given by 0 () = p =~. The location of the branch points of the spectral curve are given by a = 0 and a + ! +1. Writing (6.38) for this particular case, we nd W g (z;I) = Res F g (z 0 ;I) 2(z 0 z) p z ;z 0 = 0 ; (6.43) where the only non-zero contribution comes from a = 0. We can easily work out the rst few coecients for this example and nd W 0 (z 1 ;z 2 ) = 1 4 p z 1 p z 2 ( p z 1 + p z 2 ) 2 ; W 1 (z 1 ) = 1 32(z 1 ) 5=2 ; W 1 (z 1 ;z 2 ) = 5(z 2 1 +z 2 2 ) + 3z 1 z 2 128(z 1 ) 7=2 (z 2 ) 7=2 ; W 2 (z 1 ) = 105 2048(z 1 ) 11=2 ; W 0 (z 1 ;z 2 ;z 3 ) = 1 16(z 1 ) 3=2 (z 2 ) 3=2 (z 3 ) 3=2 ; (6.44) in agreement with the expressions given in [3]. 6.2 Method of orthogonal polynomials In this section we introduce a dierent formalism for solving a Hermitian random matrix model, called the method of orthogonal polynomials. Its main advantage with respect to the loop equations is that it allows for explicit computation of observables beyond the 1=N perturbative expansion, see [149{151] for early references and [142] for a review. After applying the method for an arbitrary matrix model, we study the double scaling limit, which enables computation of non-perturbative corrections in~ (see [138, 152] for reviews). 6.2.1 FiniteN Consider a set of polynomials P n () labeled n2N 0 and dened as P n () 1 Z n n Y j=1 Z +1 1 d( j )( 1 ;:::; n ) 2 n Y i=1 ( i ) ; d()de NV () ; (6.45) whereP 0 () = 1 andZ n is dened as the numerator inP n () but without the Q n i=1 ( i ) insertion in the integral. 7 This normalization ensures P n () is a monic polynomial with leading behavior 7 ZN is the same as the partition function of the matrix model previously dened in (6.2). 133 P n () = n +O( n1 ). For n = N the polynomial can be written as the following expectation value in the matrix model P N () =hdet(M)i : (6.46) For generaln we want to prove these polynomials form an orthogonal set. To do so, let us compute the following integral for a xed s = 0; 1;:::;n Z n Z +1 1 d( n+1 )P n ( n+1 ) s n+1 = n+1 Y j=1 Z +1 1 d( j )( 1 ;:::; n ; n+1 )( 1 ;:::; n ) s n+1 ; = 1 (n + 1) n+1 Y j=1 Z +1 1 d( j )( 1 ;:::; n ; n+1 ) " n+1 X k=1 (1) n+1k s k ( 1 ;:::; ^ k ;:::; n+1 ) # (6.47) where in the second line the factor Q n i=1 ( n+1 i ) was incorporated into one of the Vandermonde determinants. The third line is obtained by relabeling the integration variables in the k summation according to k $ n+1 and using that exchanging two rows in the determinant gives a minus sign. The notation ^ k indicates k is missing from the Vandermonde matrix. The factor between square brackets is nothing more than the following determinant det 0 B B B B B B B @ 1 1 ::: n1 1 s 1 1 2 ::: n1 2 s 2 . . . . . . . . . . . . 1 n+1 ::: n1 n+1 s n+1 1 C C C C C C C A ; (6.48) expanded along the last column. Since this determinant vanishes whenever s = 0;:::;n 1 we get the following identity Z +1 1 d( n+1 )P n ( n+1 ) s n+1 = s;n n + 1 Z n+1 Z n : (6.49) From this, the orthogonality property of the polynomials follows directly Z +1 1 d()P n ()P m () =h n n;m ; h n = 1 n + 1 Z n+1 Z n 0 ; (6.50) where h n is the norm of P n (). While (6.45) provides an explicit formula for the polynomials, in most cases we cannot write them explicitly since the analytic evaluation of the integrals is usually not possible. To deal with the polynomials it will be convenient to study the action of the multiplication by onP n (). Since 134 P n () is a polynomial of order n + 1, it can be expanded in terms of the orthogonal polynomials fP i ()g n+1 i=0 in the following way P n () =P n+1 () +S n P n () +R n P n1 () + n2 X k=0 g k P k () ; (6.51) whereR 0 = 0 and the coecient ofP n+1 () is xed to one sinceP n+1 () is monic. The coecients g k can be computed from the following identity Z +1 1 d()P m ()P n () = 0 if n>m + 1 : (6.52) This follows after using P m () is a polynomial of order m + 1, decomposing it in terms of of fP i ()g m+1 i=0 and using the orthogonality relation (6.50). From this we ndg k = 0 and the recursion relation becomes extremely simple P n () =P n+1 () +S n P n () +R n P n1 () : (6.53) As we shall see, the expectation value of all observables can be written in terms of the coecients (R n ;S n ), which depend on the particular model. When the matrix potential is evenV () =V (), the setup simplies even further sinceS n = 0. We can show this by noting (6.45) impliesP n () = (1) n P n () and applying this property to (6.53). Given a particular potential V (M), how can we determine the coecients (R n ;S n )? To ad- dress this, it is convenient to dene the functions n () obtained from a slight redenition of the orthogonal polynomials jni n () = P n () p h n e N 2 V () ; hnjmi Z +1 1 d n () m () = n;m : (6.54) We have also introduced the bra-ket notation for the Hilbert spaceH generated by these functions, that is nothing more than L 2 (R). Using (6.53), the multiplication by acts in the following way onjni jni = p R n+1 jn + 1i +S n jni + p R n jn 1i : (6.55) To write this we have been careful with the normalization, usingh n+1 =h n R n+1 which follows from computing R d()P n+1 ()P n () using (6.53) for either P n+1 () or P n (). A recursion relation for (R n ;S n ) can be derived from the following identity Z +1 1 d@ ( n () m ()) = 0 : (6.56) 135 Using that P n () = n +O( n1 ) and the orthogonality property we can evaluate the left-hand side explicitly when m =n and m =n 1, so that we nd String Equations : p R n hnjV 0 ()jn 1i = n N ; hnjV 0 ()jni = 0 (6.57) These two relations, usually called string equations, provide us with a set of recursion relations that allow us to determine the coecients (R n ;S n ). The string equations can be written for any given potential using (6.55). For the simplest quadratic potential they can be written and solved explicitly as V (M) =a 0 +a 1 M +a 2 M 2 2 =) (R n ;S n ) = 1 a 2 n N ;a 1 : (6.58) In this case the P n () are nothing more than the Hermite polynomials. Although for arbitrary potentials the string equations are quite complicated, they simplify when the potential is even, since S n = 0. In that case we only need the rst string equation in (6.57) which for a general potential V 0 () = P a 2i 2i1 gives n N =a 2 R n +a 4 R n [R n1 +R n +R n+1 ] +a 6 R n R 2 n1 +R 2 n +R 2 n+1 +2R n (R n1 +R n+1 ) +R n2 R n1 +R n1 R n+1 +R n+1 R n+2 ] +O(a 8 R 4 ) : (6.59) Each of these terms can be computed directly using (6.55) or by counting all the Motzkin paths connecting n to (n 1), where we assign the weight R i to a path connecting i to (i 1) (see appendix 3 of [151] for more details on this method). Calculating the number of terms that appear for a particular order a 2i reduces to a simple combinatorial problem which gives Number of terms for a 2i = 2i 1 i 1 : (6.60) From this we see the number of terms grows very fast with the order of the potential V (). In practice, the simplest way to computehnj 2i1 jn 1i is to write a nite dimensional matrix rep- resentation of , compute the appropriate power and extract the (n;n 1) component. Computing expectation values So far we have introduced the Hilbert spacejni = n () and shown how the action of the operator in (6.55) can be determined from the string equations (6.57). We now show how to use this 136 formalism to compute the expectation value of observables. The simplest observable is the partition functionZ N , which can be easily obtained from (6.50) Z N =N! N1 Y k=0 h k =N!h N 0 N1 Y k=1 R Nk k ; (6.61) where in the second equality we have usedh n+1 =h n R n+1 (see below (6.55)). This gives an explicit expression forZ N in terms of R n and h 0 = R d(). Extending this to the expectation value (6.6) of observables is more challenging. The central quantity we want to compute is the generating function of connected correlation functions, dened as G(~ ) = ln h he P p w=1 w TrFw (M) i i ; (6.62) where~ = ( 1 ;:::; p ) andF w (M) is an arbitrary function. Dierentiating with respect to w and evaluating at w = 0 we nd @ ~ k ~ G(~ ) ~ =0 = p Y w=1 (TrF w (M)) kw c ; @ ~ k ~ p Y w=1 @ kw w ; (6.63) where ~ k = (k 1 ;:::;k p ). The main complication in computing G(~ ) comes from the fact that theN integrals over i in the expectation value are coupled due to the Vandermonde determinant () = ( 1 ;:::; N ). To simplify this, we can write this determinant in terms of the orthonormal polynomials () = det j1 i = det P j1 ( i ) = N Y j=1 r Z N N! det P j1 ( i ) p h j1 ! : (6.64) In the second equality we have used the determinant is invariant under linear combinations of its columns to rewrite it directly in terms of the polynomials. We then multiplied each column by 1 = p h j1 =h j1 , extracted p h j1 from the determinant and used (6.61) to rewrite them in terms of the partition functionZ N . Expanding the determinant using its ordinary denition we nd 1 Z N N Y i=1 e NV ( i ) () 2 = 1 N! X ;2S N (1) + N Y i=1 (i)1 ( i ) (i)1 ( i ) ; (6.65) where is an element of the permutation group S N with parity . 8 Using this in (6.6) we can 8 The permutation group SN should not be confused with the coecient Sn appearing in the recursion relation of the orthogonal polynomials (6.53). 137 write G(~ ) as G(~ ) = ln 2 4 1 N! X ;2Sn (1) + N Y i=1 h(i) 1je P p w=1 wFw () j(i) 1i 3 5 : (6.66) We recognize the expression inside the logarithm as the determinant of an N dimensional matrix dened as A nm (~ ) =hnje P p w=1 wFw () jmi ; n;m = 0;:::; (N 1) : (6.67) In terms of this matrix we obtain the following simple expression for the generating function G(~ ) = Tr ln(A) ; (6.68) where we have used Jacobi's formula to write the determinant as a trace. Dierentiating with respect to ~ as in (6.63) we can explicitly write arbitrary correlation function. For single trace observables we nd hTrF (M)i = Tr H F ()P ; P N1 X n=0 jnihnj ; (6.69) where Tr H is the trace over the innite dimensional Hilbert spaceH =fjnig n2N 0 . We introduced the projectorP since the matrixA and trace is (6.68) not overH but over the rstN vectors. We can easily work out similar expressions for multi-trace operators h 2 Y w=1 TrF w (M)i c = Tr H F 1 ()(1P)F 2 ()P ; h 3 Y w=1 TrF w (M)i c = Tr H F 1 ()(1P) F 2 ()(1P)F 3 ()F 3 ()PF 2 () P ; (6.70) where used @ (A 1 ) =A 1 (@ A)A 1 and had to insert the projectorP whenever a product of A or its inverse appeared. These useful formulas can be used to write the expectation value of arbitrary single trace operator for nite N in terms of the recursion coecients (R n ;S n ). Some simple examples are given by Tr b 0 +b 1 M +b 2 M 2 = N1 X n=0 b 0 +b 1 S n +b 2 S 2 n +R n +R n+1 ; Tr b 0 +b 1 M Tr c 0 +c 1 M c =b 1 c 1 R N : (6.71) After solving for (R n ;S n ) from the string equation (6.57), we can insert the values and compute these expressions explicitly. For the double trace observable there is an interesting cancellation which makes the nal answer only depend on R N . 138 6.2.2 Double scaling limit Let us now describe a particular regime, called the double scaling limit, which captures certain universal features of the matrix model. One of the advantages of working in this regime is that the complicated string equations in (6.57) for (R n ;S n ) are reduced to ordinary dierential equations, making them much easier to deal with. To start, we must dene what is a critical matrix model. Consider the spectral density () in the large N limit obtained in (6.24), that we rewrite here for convenience lim N!1 h()i = 0 () = 1 2 h() p () 1 ()<0 ; (6.72) where 1 ()<0 is the indicator function and h(E) and () are polynomials that depend on the potentialV (M). The polynomial(E) only contains simple roots() = Q s i=1 (a i )(b i ) with a i andb i real. Fors = 1; 2;::: the matrix model is said to be single-cut, double-cut and so on. For our purposes, a critical potential (see section 6.5 in [153]) is dened as V (M) critical () h() = 0 ; 2[ s i=1 [a i ;b i ] : (6.73) For Hermitian matrix models, there are two types of critical models, depending on whether h() vanishes at an interior point (a i ;b i ) or precisely at one of the edges, see table 6.1. Each case is also characterized by rate at which the spectral density 0 () vanishes at the critical point. This is labeled by k2N in the third column of the same table. It turns out that the models for which h() vanishes with an odd power, are unstable. The instability can manifest as a negative spectral density (see [154] in the interior case) or an unbounded potential i.e. lim !1 V () =1, see equation (4.8) in [143] for edge critical potentials. In this section we are going to focus on the stable interior critical models, who's double scaling was rst analyzed in [155, 156] (see also [154, 157{159]). To start, let us compute the potential that results in the following large N spectral density labeled by k2N 0 () = b k 2 a 2k r a 2 2 a 2 1 [a;a] ; where b k = 2 2k+1 (k + 1)!(k 1)! a(2k 1)! ; (6.74) is a normalization constant and a2R + determines the support. Although 0 () is supported on a single interval, it vanishes at the origin 0 () 2k , meaning it corresponds to a critical model. The potential required to generate this spectral density can be easily computed by requiring the 139 Potential h() 0 () ( c ) # Stable? Non-critical potential h()6= 0 for 2[ s i=1 [a i ;b i ] - - Interior critical potential h( c ) = 0 for c 2[ s i=1 (a i ;b i ) 2k Yes 2k 1 No Edge critical potential h( c ) = 0 for c 2fa i ;b i g s i=1 2(k 1) + 1=2 Yes (2k 1) + 1=2 No Table 6.1: Classication of the dierent types of potentials in a Hermitian random matrix model. The critical potentials are characterized by the rate at which 0 () vanishes at c , with k2N. In the last column we indicate whether the system is stable or not. The highlighted row corresponds to the critical models that are the focus of this section. large N limit of the resolvent W 0 (z) in (6.23) has the correct behavior W 0 (z) = 1=z +O(1=z 2 ). Doing so, we obtain the following family of critical potentials V 0 2k () =b k k X n=0 1=2 n (1) n (=a) 2(kn)+1 : (6.75) This gives a niteN denition of critical models for which 0 () 2k . Let us stress that for xed k there is an innite number of critical potentials one could write down which produce the desired behavior 0 () 2k . We should think as V 2k () in (6.75) as a representative of this class. One of the crucial features of the double scaling limit is that the end result is universal, i.e. independent of the particular representative critical potential. In the double scaling limit we take N large while simultaneously approach the critical potential Double scaling limit : 8 > > < > > : N ! 1 V ()!V 2k () : (6.76) If these limits are taken in the right way, we shall show the string equation S k p R n hnjV 0 ()jn 1i n N = 0 ; (6.77) becomes an ordinary dierential equation which can be solved and used to perform non-perturbative computations in the double scaled model. Single k = 1 critical model Let us start with the simplest case, applying the double scaling to the k = 1 model dened from the critical potential in (6.75). In order to approach criticality we introduce the parameter and 140 Figure 6.2: Spectral density 0 (; ) for the potential in (6.78) (dashed curve). As goes to one we approach criticality and there is a phase transition from a double to a single-cut model. consider the following potential V (; ) = 1 V 2 () = 4 a 2 2 + 4 =a 2 ; (6.78) so that when ! 1 we obtain a critical model. The double scaling limit of this system was rst worked out in [155] and then rigorously studied in [159, 160]. Moreover, it was then shown in [161] that the double scaling limit is independent of the particular critical potential one chooses to perform the calculation. To get some intuition on the system away from criticality, we can compute the large N spectral density for general and nd 0 (; ) = (2=a) 4 2 8 > > < > > : 2 + (c 2 0 a 2 )=2 q c 2 0 2 ; 1 ; jj q (c 2 + 2 )( 2 c 2 ) ; 1 : (6.79) where we are omitting the indicator functions that determine the support and have dened (c 0 =a) 2 = 1 + p 1 + 3 3 ; (c =a) 2 = 1 p 2 : (6.80) Depending whether is larger or smaller than one, the system is in a single or double-cut phase respectively (see gure 6.2). Precisely at = 1 there is a phase transition which signals the criticality of the model. The string equation (6.77) can be easily obtained from (6.59) and (6.78) S k=1 =2R n +R n [R n1 +R n +R n+1 ] n N = 0 ; (6.81) where from now on we x a = 2 for convenience. The two initial conditions required to solve this 141 Figure 6.3: SolutionR n to the discrete string equationS k=1 in (6.81) for = 1 andN = 14. The red dashed line corresponds to the leading solution given in (6.83). recursion relation are given by Initial conditions : R 0 = 0 ; R 1 = h 1 h 0 = R +1 1 d 2 e NV (; ) R +1 1 de NV (; ) ; (6.82) where we can see R 0 vanishes by taking n = 0 in (6.53). Fixing a particular value of and N we can solve the integrals numerically and use the value ofR 1 to solve the recursion relation (6.81). In gure 6.3 we show an example for = 1 and N = 14. Since numerical solutions to the non-linear recursion relation are generally quite unstable, a more convenient variational method was applied in [159] (see [162] for details) that allow for stable solutions for higher values of N. For our purposes, N = 14 is enough. For small values ofn=N, the value ofR n jumps between two dierent branches, which end up merging around n=N 1. The leading behavior was rigorously computed in [160] and shown to be given by R n ' 8 > > > < > > > : 1 (1) n r 1 n N ; n N 1 ; 1 3 + 1 3 r 1 + 3 n N ; n N 1 : (6.83) We plot this analytic solution in gure 6.3, nding good agreement. We are now ready to take the double scaling limit by approaching criticality and simultaneously taking the large N limit in the following way 1 N = 1 2 ~ 2+1 ; = 1 + 2 ; (6.84) where ! 0 and (~;) are the scaling parameters associated to each of these quantities. Note the sign of is crucial, as it determines whether we approach the critical potential from either the 142 single or double-cut phase. Figuring the right power of in each of the dierent quantities is a matter of trying dierent values until one gets a useful ansatz. In the large N limit, n=N becomes a continuum variable x, related to in the following way n N = 1 +x 2 : (6.85) It is important to mention that we allowx to take any real value, including innite. In this way, we can approach whatever value ofn we desire, not necessarilynN. The only thing we are missing to compute the double scaling limit of the string equation (6.81) is an ansatz for R n . Building on the numerical solution in gure 6.3 and the leading behavior (6.83), let us consider R n = 1 (1) n r(x) + 2k+1 X i=2 [f i (x) + (1) n g i (x)] i : (6.86) Whiler(x) controls leading scaling behavior, the functionsff i (x);g i (x)g determine the subleading contributions with and without the (1) n insertion. Putting everything together, we can expand the string equation (6.81) in a power series in and obtain S k=1 = 4f 2 (x) (x +r(x) 2 ) 2 + [4f 3 (x) 2g 2 (x)r(x)] + + (1) n 2 4f 2 (x)r(x) 1 2 ~ 2 r 00 (x) 3 +O( 4 ) : (6.87) For the functionsff 2 (x);g 2 (x)g we get algebraic equations that can be easily solved. This is not the case of r(x), which instead satised the following dierential equation, that is nothing more than Painleve II lim !0 1 3 S k=1 = 0 () r(x) 3 1 2 ~ 2 r 00 (x) +r(x)x = 0 : (6.88) Thus, we have shown that in the double scaling limit the string equation for the recursion coecients R n is reduced to a ordinary dierential equation for r(x). The boundary conditions are obtained by matching with the leading solution (6.83), so that we nd Boundary conditions : lim x!1 r(x) = p x lim x!+1 r(x) = 0 : (6.89) 143 Single k critical model We can apply the same procedure to any of the critical potentials in (6.75) labeled by k2N. To do so, we must slightly modify the scaling of (N; ;n) in (6.84) and (6.85), and consider instead 1 N = 1 2 ~c 2k 2k+1 ; = 1 +c 2k 2k ; n N = 1 +c 2k x 2k ; (6.90) wherec 2k is a normalization coecient that is conveniently chosen for each value of k. The ansatz for R n in (6.86) remains the same. We now proceed in the same way as the k = 1 and writeS k (6.77) in a series expansion in . The rst two orders give S k = k+1 X i=1 2i 1 i 1 a 2i 1 + (1) n+1 r(x) k+1 X i=1 2i 2 i 1 a 2i +O( 2 ) ; (6.91) wherea 2i are the coecients of the critical potential (6.21) dened asV 0 2k () = P k i=1 a 2i 2i1 . Both of these terms can be written in terms of a hypergeometric function and vanish when a (dened in (6.75)) is set to a = 2. Setting to zero the higher order contributions ofS k gives a set of algebraic equations forff i (x);g i (x)g. The rst non algebraic constraint appears at order 2k+1 multiplied by (1) n , giving an ordinary dierential equation for r(x) lim !0 1 2k+1 S k = 0 () K 2k +r(x)x = 0 ; (6.92) where K 2k is a polynomial in r(x) and its derivatives. For the rst few values of k they can be computed explicitly and written as K 2 =r(x) 3 1 2 ~ 2 r 00 (x) ; K 4 =r(x) 5 5 6 ~ 2 r(x) r(x) 2 00 + 1 6 ~ 4 r (4) (x) ; K 6 =r(x) 7 7 6 ~ 2 r(x) 2 r(x) 3 00 7 10 ~ 4 h 5r 0 (x) 2 r 00 (x) + 3r(x)r 00 (x) 2 + 4r(x)r 0 (x)r (3) (x) +r(x) 2 r (4) (x) i 1 20 ~ 6 r (6) (x) ; K 8 =r(x) 9 3 2 ~ 2 r(x) 3 (r(x) 4 ) 00 + ( )~ 4 + ( )~ 6 + 1 70 ~ 8 r (8) (x) ; (6.93) where we have omitted some of the terms in K 8 and for each case we xed c 2k in (6.90) to c 2k = (1; 3=8; 1=8; 5=128) so that the leading term is given by r(x) 2k+1 . From this expression we can extrapolate the behavior of K 2k for general k K 2k =r(x) 2k+1 1 6 (2k + 1)~ 2 r(x) k1 r(x) k 00 + + (1) k k!(k 1)! 2(2k 1)! ~ 2k r (2k) (x) : (6.94) 144 Moreover, we note this generalk behavior and the particular expressions in (6.93) can be computed from the following recursion relation K 2k = 2k 2k 1 r(x) Z d xr( x)K 0 2(k1) 1 4 ~ 2 K 00 2(k1) : (6.95) This corresponds to the Painleve II hierarchy of ordinary dierential equations, related to the modied Korteweg{de Vries (mKdV) integrable hierarchy. Similarly as in the k = 1 case in (6.89), the boundary conditions are given by Boundary conditions : lim x!1 r(x) = (x) 1=k lim x!+1 r(x) = 0 : (6.96) Multiple critical models Let us now consider the k critical model and perturb it using the other critical models with i<k. To do so, we need to take a superposition of their respective critical potentials (6.75) V (; ) = 1 " V 2k () + k1 X i=1 c 2k c 2i t 2i 2(ki) V 2i () # : (6.97) The perturbation away from the k critical model is parametrized by the parametersft 2i g k1 i=1 . We have appropriately included factors of in order to yield an appropriate result in the double scaling limit. To accommodate for t 2i 6= 0, the ansatz for and n is slightly modied compared to (6.90), see table 6.2. Taking the double scaling limit of the string equationS k (6.77) the same way as before, we nd lim !0 1 2k+1 S k = 0 () K 2k + k1 X i=1 t 2i K 2i +r(x)x = 0 : (6.98) The string equation now gets contributions from the dierential polynomials K 2i with i k. Formally taking the k! +1 we nd String equation : 1 X i=1 t 2i K 2i +r(x)x = 0 (6.99) This is the most general string equation obtained from the stable and interior critical potentials. A particular model is specied by xing the value of the coecients t 2i that control the inclusion of each critical model. 145 Matrix model Scaling parameter Double scaling limit ! 0 N ~ 1 N = 1 2 ~c 2k 2k+1 = 1 + k1 X i=1 c 2k c 2i t 2i 2(ki) +c 2k 2k n x n N = 1 + k1 X i=1 c 2k c 2i t 2i 2(ki) +c 2k x 2k R n r(x) R n = 1 (1) n r(x) + 2k+1 X i=2 [f i (x) + (1) n g i (x)] i = 2n () f + (x;) 2n () = (1) n p ~=2f + (x;) 2n+1 () f (x;) 2n+1 () = (1) n p ~=2f (x +~ =2;) Table 6.2: Ansatz for the dierent parameters in the matrix model involving the double scaling of a critical model with potential V 2k () (6.21) perturbed by other critical models with i<k (see (6.97) for the full potential). The perturbation away from the k model is parametrized by the coecientsft 2i g k1 i=1 . The double scaling limit! 0 of the string equationS k (6.98) can be worked out using the rst four rows of this table. The normalization coecient c 2k is xed so that when double scaling a single critical model we have K 2k =r(x) 2k+1 +O(~ 2 ). For the rst few values we have c 2k = (1; 3=8; 1=8; 5=128). Computing expectation values So far we have shown how the string equationS k (6.77) becomes an ordinary dierential equation for r(x) in the double scaling limit. Let us now study the expectation value of observables. To start, we \zoom in" 0 by rescaling =. Same as for the recursion coecients R n , to take the double scaling of theL 2 (R) functions n () we must distinguish between even and oddn. This gives two independent functionsf (x;), see last row of table 6.2. In terms of the bra-ket notation of (6.54) we now have jni = n () ! double scaling jx;si =f s (x;) ; s = : (6.100) The countable statesjni are replaced by the uncountablejxi with x2R together with an extra spin degree of freedom. The orthonormality ofjni becomes hs;xjx 0 ;s 0 i = Z +1 1 df s (x;)f s 0(x 0 ;) =(xx 0 ) s;s 0 : (6.101) 146 The Dirac delta arises from n;m = 1 2 Z de i(nm) = 1 2 Z de i(xy) 2 ~ = (~)(xy) ; (6.102) where in the last step we have changed the integration variable to 2=~ and used the usual integral representation of the Dirac delta. Consider the expectation value of a single trace observable, which before any double scaling is computed from (6.69), explicitly given by hTrF (M)i = N1 X n=0 hnjF ()jni = N1 X n=0 Z +1 1 d n ()F () n () : (6.103) Splitting the sum between even and odd values in n and using the scaling behavior in each case we nd hTrF (M)i = X s= Z 1 dxhs;xjF ()jx;si : (6.104) where we have used ~ 2 P n ! R dx together with x(n) with x(0) =1 and x(N 1) = . 9 To compute this, all we need is the action of on the statesjx;si. Applying the double scaling to (6.55) we nd jx;si = s~@ x +r(x) jx;si : (6.105) Using this in (6.104) we nd lim !0 1 2m hTrM 2m i = X s= Z 1 dxhxj [(s~@ x +r(x))(s~@ x +r(x))] m jxi ; (6.106) together withhTrM 2m+1 i = 0, which is ultimately expected since the potential dening the matrix model is even. All the even powers of M are nicely combined in the following observable hTre `M 2 i = X s= Z 1 dxhxje Hs jxi ; (6.107) where we have dened H s =(~@ x ) 2 + r(x) 2 s~r 0 (x) ; ` == 2 : (6.108) Dierentiating (6.107) with respect to ` and setting ` = 0 we obtain (6.106) for arbitrary m. The explicit formula in (6.107) allows us to exactly compute observables in the double scaled model to 9 We are using n =~=2. 147 all order in~, including both perturbative and non-perturbative contributions. The right-hand side involves a trace in a one dimensional quantum mechanical system, characterized by the Hamiltonian H s . An analogous computation for the double and triple trace observables in (6.70) gives h 2 Y i=1 Tre ` i M 2 i c = X s= Z 1 dx 1 Z +1 dx 2 hx 1 je 1 Hs jx 2 ihx 2 je 2 Hs jx 1 i ; h 3 Y i=1 Tre ` i M 2 i c = X s= Z 1 dx 1 Z +1 dx 2 hx 1 je 1 Hs jx 2 i Z +1 dx 3 hx 2 je 2 Hs jx 3 ihx 3 je 3 Hs jx 1 i Z 1 dx 3 hx 2 je 3 Hs jx 3 ihx 3 je 2 Hs jx 1 i : (6.109) Note each spin sector in the quantum mechanical model are decoupled. This follows after noting (6.105) implieshs;xj 2w jx 0 ;si = 0. Although the expectation value of higher trace operators become increasingly complicated, it is clear they are all computed from the matrix elements of the dierential operatorH s . In fact, they can all be written in terms of the following object, called the self-reproducing kernel L s (E 1 ;E 2 ) = Z 1 dx E 1 ;s (x) E 2 ;s (x) ; (6.110) where E;s (x) are the eigenfunctions ofH s H s E;s (x) =E E;s (x) : (6.111) The kernel can exactly integrated in the following way (E 1 E 2 )L s (E 1 ;E 2 ) = Z 1 dx [H s E 1 ;s (x)] E 2 ;s (x) E 1 ;s (x) H s E 2 ;s (x) =~ 2 Z 1 dx@ x E 1 ;s (x) 0 E 2 ;s (x) 0 E 1 ;s (x) E 2 ;s (x) =~ 2 E 1 ;s (x) 0 E 2 ;s (x) 0 E 1 ;s (x) E 2 ;s (x) x= ; (6.112) where in the rst equality we used the denition of E;s (x) (6.111). The boundary term atx =1 does not contribute given that the potential inH s (6.108) goes to plus innity due to the boundary condition (6.96) of the string equation (meaning E;s (x) vanishes in the limit). Putting everything together the self-reproducing kernel can be written as L s (E 1 ;E 2 ) =~ 2 " E 1 ;s (x) 0 E 2 ;s (x) 0 E 1 ;s (x) E 2 ;s (x) E 1 E 2 # x= ; (6.113) 148 which is sometimes called the Christoel-Darboux formula. 10 Its diagonal elements are given by L s (E;E) =~ 2 (@ E E;s ) (@ x E;s ) E;s (@ E @ x E;s ) x= ; (6.114) This is very much related to the expectation value of the spectral density () = Tr(M). In fact, starting from (6.107) and (6.109) we can write simple relations between the self-reproducing kernel and correlation functions involving () insertions h()i =jj X s= L s ( 2 ; 2 ) ; h( 1 )( 2 )i c =j 1 2 j X s= L s ( 2 1 ; 2 1 )( 2 1 2 2 )jL s ( 2 1 ; 2 2 )j 2 ; (6.115) where the factors ofj i j come from the Jacobian of a change of variables required to go from Tr(M) to Tre `M 2 . 10 This relation can also be written for the functions n() built from the orthogonal polynomials, before any double scaling limit. 149 Chapter 7 Two dimensional quantum gravity In this chapter we show how the matrix models we have introduced in the previous chapter can be used to compute Euclidean partition functions of certain two dimensional quantum gravity theories. One of the main advantages of this method is that in some cases the matrix models can supply a stable and fully non-perturbative denition of the gravity theory. In particular, we explicitly show how (doubly) non-perturbative eects can be reliable computed through the matrix model. 7.1 Jackiw-Teitelboim gravity Let us start by recalling some general aspects of two-dimensional gravity, see [163, 164] for useful references. In two space-time dimensions the most general scalar curvature invariant is built from the Ricci scalarR and contractions of its covariant derivatives, e.g. (rR) 2 = (r R) (r R). This is because for any metric, the Riemann and Ricci tensor are xed byR according to R = 1 2 R (g g g g ) ; R = R 2 g : (7.1) The most general theory of gravity is constructed from the following action I[g ] = Z d 2 x p gL(R;r R;r r R;::: ) : Perhaps the most natural theory is obtained by takingL =R, corresponding to the Einstein-Hilbert action in higher dimensions. However, using the second relation in (7.1), the equations of motion of this theory are satised by all metrics. This means that to formulate a non-trivial theory of two 150 dimensional pure gravity, we must necessarily include higher curvature terms in the Lagrangian. A particular class of theories is obtained by taking a Lagrangian that is only a function of the Ricci scalarR I[g ] = Z d 2 x p gf(R) ; (7.2) wheref(R) is a smooth function. One disadvantage of working with this class of theories is that the equations of motion are at least fourth order in the metric, sinceR already contains two derivatives of the metric and f(R) = 0 + 1 R is trivial. The simplest non-trivial case is given by I[g ] = Z d 2 x p g 0 + 2 R 2 : (7.3) To avoid the fourth order dierential equations it is convenient to intregrate in an auxiliary scalar eld (x ), called the dilaton from now on, I[g ; ] = Z d 2 x p g 0 + R 2 4 2 : (7.4) This action has second order dierential equations. Varying with respect to we get an algebraic equation that is solved by = 0 = 2 2 R so that replacing back into the action I[g ; 0 ] we recover the previous purely gravitational action in (7.3). Altogether, it means both theories in (7.3) and (6.26) describe the same dynamics, at least classically. We should not think of (x ) as a matter eld, but instead as a scalar that describes a gravitational degree of freedom. A similar story holds for more general gravity theories given by (7.2) by considering the following Einstein-dilaton action I[g ; ] = Z d 2 x p g RV () : (7.5) Varying with respect to givesR =V 0 (). If the second derivative of V () is non vanishing, we can invert this relation substitute back in the action and obtain a purely gravitational theory for some function f(R) (which is determined by the potential V ()). In practice it is much simpler to work with the dilaton gravity theory in (7.5). There are however more general dilaton gravity actions one can write down. The most general action with second order dierential equations is given by I[~ g ;'] = Z d 2 x p ~ g h U(') e RW (') +Z(') (@ ') (@ ') i ; (7.6) 151 where U('), W (') and Z(') are arbitrary functions. This theory can be simplied by redening the metric in the following way g =e 2(') ~ g ; (') = 1 2 Z '(x ) Z(y) U 0 (y) dy : (7.7) Using that the Ricci scalar in two dimensions transforms as e 2 e R =R + 2r 2 and integrating by parts, the additional contribution cancels the kinetic term for . Further redening the dilaton eld as U(') = and assuming U(') is an invertible function, we arrive at (7.5) with V () = e 2(') W ('). This shows it is enough to consider dilaton gravity theories dened from the action in (7.5), whose equations of motion are given by I = 0 =) R =V 0 () ; g I = 0 =) r r = g 2 V () : (7.8) Let us now analyze the solutions to these equations. Given a root 0 of the potential V ( 0 ) = 0, we can immediately construct a solution with constant dilaton (x ) = 0 and curva- tureR =V 0 ( 0 ). The value of the cosmological constant is set by the derivative of the potential at 0 R =V 0 ( 0 ) = 2 : (7.9) Since the Riemann tensor (7.1) is entirely determined byR, depending on the value of this corresponds to an anti-de Sitter, de Sitter or at space solution. It is interesting to perturb away from this maximally symmetric solutions by expanding the potential away from 0 V () =V 0 ( 0 )( 0 ) +O( 0 ) 2 ; (7.10) so that the corresponding action (7.5) becomes I[g ; ] = 0 Z d 2 x p gR + Z d 2 x p g( 0 ) (R 2) +O( 0 ) 2 : (7.11) The Jackiw-Teitelboim gravity theory is dened by dropping the subleading corrections and dening the dilaton = 2( 0 ) I JT [g ;] = 0 Z d 2 x p gR + 1 2 Z d 2 x p g (R 2) (7.12) This action captures the rst order perturbation away from the maximally symmetric solutions of any two dimensional dilaton gravity theory. It is fully determined by the two parameters 0 and 152 . While the Einstein term multiplied by 0 does not contribute to the equations of motion, the variation of the second term gives R = 2 ; r r + g = 0 : (7.13) From this we observe the metric is still xed to the maximally symmetric solution. The perturbation captured by JT gravity is instead given by the prole of the dilaton eld , that we should think of as describing a gravitational degree of freedom. From now on let us focus on the case of negative cosmological constant and x our units by requiring =1, so that JT gravity captures perturbations away from AdS 2 . This is perhaps the most interesting case. As shown in section 3.1, the near horizon limit of an extremal Reiss- ner{Nordstr om black hole in four dimensional Einstein-Maxwell theory gives an AdS 2 S 2 metric. By performing a dimensional reduction in the AdS 2 sector, four dimensional Einstein-Maxwell be- comes a dilaton gravity theory of the form given in (7.5) which becomes JT gravity in the limit = 0 1 (see [165] for a recent discussion and references therein). As such, JT gravity captures the dynamics of higher dimensional near extremal black holes. From this perspective the constant 0 is xed by 0 = S 0 2 , where S 0 is the entropy of the four dimensional extremal black hole. Before analyzing quantum aspects of JT gravity, let us explicitly write the solution to the equations of motion (7.13) ds 2 =(r 2 1)dt 2 + dr 2 r 2 1 ; (r) = h r ; (7.14) where r 1 and t2R. These coordinates describe a patch of global AdS 2 , with the boundary at r 1. The metric contains a horizon at r = 1 analogous to the Rindler horizon in Minkowski, see the left diagram in gure 7.1 for the Penrose diagram. 1 7.1.1 Disc partition function Let us now discuss the quantization of JT gravity in Euclidean signature (see [166] for the Lorentzian case). To start let us consider the partition function that includes the contribution from surfaces 1 The location of the horizon r = 1 can be modied to r =r h by a change of coordinates (t;r)! (r h t;r=r h ). 153 AdS Boundary AdS Boundary Figure 7.1: The left diagram corresponds to the Penrose diagram of Lorentzian AdS 2 , with the shaded region corresponding to the region covered by the (t;r) coordinates in (7.14). On the right, we have Euclidean AdS 2 (the Poincare disc) as described by the coordinates (;r) in (7.21). The boundary curve x (u) (7.22) determines the location of the boundary @M where the boundary Dirichlet boundary conditions (7.16) are specied. with a disc topology Z 0 Z Disc Dg D exp I E JT [g ;] ; (7.15) where I E JT [g ;] is the Euclidean action of JT gravity. To properly dene the partition function we must determine the boundary conditions for the metric and dilaton. We consider the following Dirichlet conditions, xing the metric and dilaton to constant values at the boundary @M ds 2 @M =g uu du 2 = du 2 2 ; (x ) @M = b ; (7.16) where is a small parameter and u (the coordinate along @M) is dened such that g uu = 1= 2 . Moreover, by shifting (;u)! b (;u) we can x b arbitrarily, which for convenience we take as b = 1=2 from now on. 2 Although we have xed both the metric and dilaton to a constant values at the boundary, it seems we have a single parameter . There is however another parameter, , that determines the range of the boundary coordinate u2 [0;]. It is closely related to the length of the boundary L @M L @M = Z 0 du p g uu = : (7.17) The boundary conditions are therefore parametrized by (;). 2 One could consider the more general case in which b = b (u) is a function of the boundary coordinate u. We shall not explore this possibility here. 154 Having xed the boundary conditions, the action has to be supplemented with the appropriate boundary terms that yield a well dened variational problem (see [167] for a general analysis). The full Euclidean JT gravity action can be written as I E JT [g ;] =S 0 (M) 1 2 Z M d 2 x p g(R 2) Z @M du p g uu (K 1) ; (7.18) where K is the extrinsic curvature of @M and we have identied the Euler characteristic ofM as (M) = 1 4 Z M d 2 x p gR + 1 2 Z @M du p g uu K = 2(1g)n ; (7.19) with g and n the genus and number of boundaries ofM. Since in this case we are restricting ourselves to the disc topology, we have (M) = 2 1 = 1. Putting everything together the disc partition function (7.15) is given by Z 0 (;) =e S 0 Z Disc Dg Z D exp 1 2 Z M d 2 x p g(R 2) + 1 2 Z @M du (K 1) 2 : (7.20) Since the dilaton only appears linearly in the action, its path integral is particularly simple. If we take the integration contour in the complex direction, the integralD gives a Dirac delta that xes R =2. This is the crucial step which makes the path integral over metrics tractable, as there is a single bulk metric withR =2 and disc topology, corresponding to the analytic continuation of (7.14) ds 2 = (r 2 1)d 2 + dr 2 r 2 1 ; + 2 ; (7.21) where the periodicity in is xed such that the metric is smooth as r! 1. This metric describes the Poincare disc, see right diagram in gure 7.1. While it corresponds to the unique metric with constant negative curvature and disc topology, there is still an innite number of degrees of freedom associated to the location of the boundary@M where we x the boundary conditions (7.16). These can be parametrized by a curve in the Poincare disc parametrized by u x (u) = ((u);r(u)) ; u2 [0;) ; (7.22) where determines the range of the boundary coordinate u. The functions (u) and r(u) are not independent but related by the boundary condition of the metric (7.16) according to g uu (u) = (r(u) 2 1) 0 (u) 2 + r 0 (u) 2 r(u) 2 1 = 1 2 (7.23) 155 All the possible boundary conditions are thus parametrized by a single function (u) which must satisfy the periodicity condition (u +) = (u) + 2 when going around the Poincare disc (see right diagram in gure 7.1). The path integral in (7.20) is simplied to Z 0 (;) =e S 0 Z D(u) exp 1 2 Z 0 du K[(u)] 1 2 : (7.24) While evaluating this path integral for nite is very challenging [168, 169], it greatly simplies when ! 0. To start, we can use (7.24) to solve for r(u) in terms of (u) in a series in r(u) = 1 0 (u) +O() : (7.25) This rst order contribution turns out being enough for our purposes. From this we see the small limit corresponds to taking the boundary@M close to the boundary of the Poincare disc. To write the exponent in (7.24), we need the extrinsic curvature K, dened as K =r n = n g uu d 2 x du 2 + dx du dx du ; (7.26) where is the connection of the Poincare disc (7.21) and n the unit normal vector to the boundary. 3 It can be easily computed by requiring that it is perpendicular to the tangent vector T = ( 0 (u);r 0 (u)) and then normalizing n (u) = r 0 (u); 0 (u)f(u) 2 f(u) p 0 (u) 2 f(u) +r 0 (u) 2 =f(u) ; where f(u) =r(u) 2 1 : Using this in (7.26) together with (7.25), it is straightforward to compute K in a perturbative expansion in and nd K = 1 + 2 0 (u) 000 (u) 3 00 (u) 2 2 0 (u) 2 + 1 2 0 (u) 2 2 +O( 4 ) ; where we identied the rst term between square brackets as the Schwartzian derivative of the function (u) Sch(;u) = 000 (u) 0 (u) 3 2 00 (u) 0 (u) 2 : (7.27) Using this in (7.24) the disc partition function in the ! 0 limit becomes Z 0 () lim !0 Z 0 (;) =e S 0 Z D(u) exp 1 2 Z 0 du Sch(;u) + 1 2 0 (u) 2 : (7.28) 3 See equation (10.13) in [170] for the second explicit expression of K. 156 The gravitational path integral of JT gravity in the disc has been reduced to the path integral of a quantum mechanical system (usually called the \Schwarzian quantum mechanics") characterized by the function (u)2 Di(S 1 ). Although in principle it seems the path integral is over the whole group Di(S 1 ), one has to mod out by SL(2;R). We can see this by noting SL(2;R) correspond to zero modes of the action of exponent, which becomes clear after rewriting in the following way Sch(;u) + 1 2 0 (u) = Sch (tan(=2);u) ; (7.29) and using the Schwarzian derivative is by invariant under SL(2;R) transformations of tan(=2). From the gravity perspective this invariance comes from the SL(2;R) isometries of the Poincare disc (7.21) which transform equivalent pathsx (u) determining the boundary condition (7.22) (see gure 7.1). Taking this into account, we can write (7.28) more precisely as Z 0 () =e S 0 Z 2G De I Sch [(u)] ; G = Di(S 1 )=SL(2;R) ; (7.30) where we have dened the action of the Schwartzian quantum mechanics as I Sch [(u)] = 1 2 Z 0 du Sch(;u) + 1 2 0 (u) 2 : (7.31) One of the remarkable features of this path integral can be evaluated exactly. This follows by introducing some fermionic degrees of freedom, using supersymmetric localization and proving the integral is one-loop exact [171]. 4 Instead of proving this directly, let us instead evaluate Z 0 () in the one-loop approximation which is in fact the exact result. To do so, we note (u) = 2 u is a saddle point of the action I Sch [(u)]. The perturbation away from the saddle can be parametrized by a periodic function "(u) of period , dened from (u) = 2 u +"(u) ; "(u) = X n2Z " n e iu 2n ; (7.32) where we have expanded "(u) in a Fourier series with " n =" n . The rst terms in the expansion of I Sch [(u)] are given by I Sch [(u)] = 2 1 4 Z 0 du " 0 (u) 2 2 2 " 00 (u) 2 +O(" 4 ) ; = 2 + 2 X n2Z n 2 n 2 1 j" n j 2 +O(" 4 n ) ; (7.33) 4 As explained in [171] the one-loop exactness of the partition function follows from the Duistermaat-Heckman formula [172]. 157 where we dropped total derivatives. From this expression we observe there are three zero modes n = 0;1. These corresponds to the three SL(2;R) transformations and must therefore be excluded from the Fourier expansion of "(u). The only thing we are missing to compute the path integral is the measureD. Following [171], we note the integral is overG = Di(S 1 )=SL(2;R), which is a symplectic manifold. There is a natural measure in the spaceG that is obtained from the symplectic form , which to quadratic order in "(u) is given by 5 = 1 2 Z 0 du 2 2 d" 0 (u)^d" 00 (u)d"(u)^d" 0 (u) +O(" 3 ) = 4 X n2 n(1n 2 )dRe(" n )^ dIm(" n ) +O(" 3 n ) ; (7.34) where in the second equality we have inserted the Fourier expansion (7.32) (omitting the zero modes n = 0;1). The symplectic form is a closed and non-degenerate 2-form onG, where d is an exterior derivative that acts on (u) or "(u) but not on the coordinate u (see [171] for more details). The measure D is taken in the usual way, as the square root of the determinant of . Putting everything together, the disc partition function (7.30) is given by Z 0 () =e S 0 + 2 = Y n2 4n(n 2 1) Z +1 1 dRe(" n )dIm(" n )e 2 2 n 2 (n 2 1)(Re("n) 2 +Im("n) 2 ) ; =e S 0 e 2 = Y n2 2 n : (7.35) The nal product can be rewritten in terms of the Riemann zeta function (s) as Z 0 () =e S 0 e 2 = exp @ s X n2 n 2 s s=0 =e S 0 e 2 = exp @ s (s) 1 (2) s s=0 : (7.36) The derivative can be evaluated exactly and we arrive at the nal result for the disc partition function of JT gravity Z 0 () = e S 0 2 p 4 e 2 = =) 0 (E) = e S 0 4 2 sinh(2 p E) ; (7.37) where we have also written the corresponding disc spectral density 0 (E), dened from the Laplace transform of the partition function Z 0 () = Z +1 0 dE 0 (E)e E : (7.38) 5 This expression is obtained from equation (114) of [3], after taking = 1 and accounting for the dierent conventions in the denition of"(u). The general form of beyond quadratic order is given in equation (2.8) of [171]. 158 Figure 7.2: Topological expansion developed in [3] for the Euclidean partition function of JT gravity with a single boundary of renormalized length indicated in green. Let us reiterate that although this comes from a one-loop computation, the result of [171] shows it is actually the exact answer, including all quantum contributions. 7.1.2 Topological expansion and random matrices The computation of the previous subsection only considered contributions from metrics with the disc topology. Due to the Euler characteristic(M) in the Euclidean action (7.18), higher genus surfaces are suppressed by factors of e S 0 . We can still incorporate those contributions by proceeding in the same way, solving rst the path integral over the dilaton which localizes the remaining path integral over metrics with constant negative curvatureR =2. In this way, the general topological expansion can be written as Z( 1 ;:::; n ) = 1 X g=0 e S 0 2(g1)+n Z g ( 1 ;:::; n ) ; (7.39) where we have allowed for n-asymptotic boundaries, each with the boundary conditions specied by i according to (7.16). 6 The partition function at xed genus Z g ( 1 ;:::; n ) is given by Z g ( 1 ;:::; n ) = Z Mg;n Dg exp " 1 2 Z @Mg;n du (K 1) 2 # ; (7.40) whereM g;n corresponds to the set of two dimensional metrics with constant negative curvature, genus g and n boundaries. Quite remarkably, this quantity was evaluated exactly in [3], building on a recursion relation derived by Mirzakhani for the volume of the moduli space of hyperbolic Riemann surfaces [173]. Let us describe the general procedure that enables the computation. The decomposition of the 6 From now on, we are always implicitly taking the ! 0 limit. 159 surfaces developed in [3] for the rst few terms in the topological expansion of the single boundary partition function Z() is shown in gure 7.2. The rst contribution is the disc partition function we have already computed exactly in (7.37). The calculation coming from the genus one surface is done in two steps, by splitting the surface along a geodesic of length b, indicated in gure 7.2 by the blue interior boundary. To the left of the geodesic, we are left with a \trumpet" geometry, whose partition function involves the extrinsic curvature K in (7.40) and can be computed exactly in a very similar way to the disc topology. The end result is given by [3] Z Trumpet (;b) = e b 2 =4 2 p : (7.41) The contribution from the surface to the right of the geodesic does not involve any contribution from the exponential in (7.40) which is only integrated along the boundary @M g;n . Therefore, it only involves an integral over the moduli of genus one surfaces with constant negative curvature and a xed geodesic boundaryb. These types of integrals, called the Weyl-Petersson volumes, have been well studied and computed by Mirzakhani for an arbitrary number of geodesic boundaries and genus. Denoting them V g;n (b 1 ;:::;b n ), the Weyl-Petersson volume is a polynomial in b i that is computed from a closed recursion relation [173]. 7 The rst few Weyl-Petersson volumes for a single geodesic boundary are given by V 1;1 (b) = (b 2 + 4 2 ) 48 ; V 2;1 (b) = (b 2 + 4 2 )(b 2 + 12 2 )(5b 4 + 384 2 b 2 + 6960 4 ) 2211840 ; (7.42) see appendix B of [174] for more explicit expressions. Using this together with the trumpet partition function, we obtain Z g () by integrating over all the possible values of b Z g () = Z 1 0 dbbZ Trumpet (;b)V g;1 (b) : (7.43) The measure of integration dbb comes from integrating over the twist when gluing the trumpet to the rest of the surface [3]. In this way, we can explicitly compute the topological expansion of the partition functionZ() to whatever order we desire. The generalization to an arbitrary number of asymptotic boundaries (7.39) is given by 8 Z g ( 1 ;:::; n ) = n Y i=1 Z 1 0 db i b i Z Trumpet ( i ;b i )V g;n (b 1 ;:::;b n ) : (7.44) 7 For the explicit recursion relation see [174] or equation (D.11) in [146], where T (b;b 0 ;b k ) is given (D.20) and D(b;b 0 ;b 00 ) is obtained from (D.6). 8 For g = 0 and n = 2 the corresponding volume is formally dened as V0;2(b1;b2) =(b1b2)=b1. 160 Another realization of [3] was the fact that this topological expansion coincides with the one obtained from the loop equations of a particular double scaled Hermitian matrix model. In fact, the connection between Mirzakhani's recursion relation and the matrix model expansion had been previously discovered by Eynard and Orantin [175, 176]. The appropriate spectral curvey(z) (6.21) that provides a perturbative denition of the matrix model is obtained from the disc partition function (7.37) which, from the matrix model perspective, corresponds to the leading expectation value of the eigenvalue spectral density (6.8). Using (6.27) we nd 0 () = 1 4 2 sinh(2 p ) =) y(z) = 1 4 sin(2 p z) : (7.45) Fixing the spectral curve in this way, the topological expansion of the gravity partitionZ( 1 ;:::; n ) was matched in [3] to the expectation value of the following observable in the matrix model Z( 1 ;:::; n )' n Y i=1 Tre i M c (7.46) whereM is the random Hermitian matrix ande S 0 =~ as dened in 6.2. This shows the equivalence between the topological expansion in JT quantum gravity and a Hermitian randon matrix model to all orders in the topological expansion. 9 The symbol' in (7.46) indicates the matching holds up to non-perturbative corrections ine S 0 and~ respectively. From the JT gravity perspective, these are actually doubly non-perturbative eectsO(e e S 0 ), and it us currently an open question how to compute them from the gravitational path integral. On the other hand, non-perturbative contributions in the matrix modelO(e 1=~ ) are much simpler to deal with, for instance using the method of orthogonal polynomials described in section 6.2. This suggests the random matrix model might supply a non-unique but fully non- perturbative denition of JT quantum gravity. The rst step in accounting for non-perturbative eects in the matrix model is deriving the appropriate potential which when double scaling gives rise to the leading eigenvalues spectral density 0 () in (7.45). To do so, let us expand it in a series in 0 () = 1 X n=0 (2) 2n1 (2n + 1)! n+1=2 = 1 2 1=2 + 3 3=2 + 3 15 5=2 +O( 7=2 ) : (7.47) 9 As discussed in subsection 6.2.2, the expectation value of matrix model observables in the double scaling limit actually involve the rescaled matrix M= 2 , where ! 0 (see for example equations (6.107) and (6.108)). From the gravity perspective taking small is analogous to the ! 0 in (7.28), which essentially means the boundary conditions at @M are close to the boundary of the Poincare disc, see (7.25) and gure 7.1. The connection between the nite cut-o partition function in JT gravity [168, 169] and the matrix model was recently investigated in [8]. 161 From table 6.1 with c = 0, we know each of these terms in the expansion can be obtained by taking the double scaling limit of an edge critical potential (last two rows). One way to generate each of these terms is to start from the following large N spectral density lim N!1 h()i = c n 2 a 2n r (a) a 2 1 [0;a] ; c n = 2 2n+3 n!(n + 2)! a(2n + 1)! : (7.48) For small , these spectral densities have the appropriate behavior corresponding to each term in the expansion of 0 () in (7.47). The critical potential required for a given value of n can be easily worked out by requiring the resolvent in (6.23) has the correct large z behavior W 0 (z) = 1=z +O(1=z 2 ). The nal result gives V n () =c n n+1 X m=0 1=2 m a(1) m n + 2m (=a) n+2m = c n a n + 2 (=a) n+2 +O(=a) n+1 : (7.49) We immediately identify an issue, given that for odd values of n the potential is unstable, i.e. V 2n+1 (1) =1. This is the stability issue was indicated for some of the double scaled models in table 6.1. While the double scaled model can be dened in perturbation theory for any value ofn, only forn even the model is non-perturbatively well dened. Since the matrix model relevant for JT gravity contains both even and odd values ofn (7.47), the Hermitian matrix model dened through (7.45) is ultimately non-perturbatively unstable and cannot be used to provide a non-perturbative denition of JT gravity. 10 Although from a dierent perspective, this issue was already recognized in [3], see their section 5. It is an interesting question whether it is possible to construct a dierent matrix model that has the same perturbative behavior that ensures the matching in (7.46) with the JT gravity partition functions, but is non-perturbatively stable. A proposal using complex instead of Hermitian matrix models was given in [177], while preliminary investigations using normal matrices was carried out in [178]. 7.2 N = 1 Jackiw-Teitelboim supergravity In this section we perform a similar construction but for a supersymmetric version of Jackiw- Teitelboim gravity. Our motivation is not only to nd other solvable theories of two-dimensional 10 The superposition between dierent critical potentials must be taken rather carefully, as discussed towards the end of subsection 6.2.2. In particular see equation (6.99), where the coecients t2i control each critical potential. 162 quantum gravity, but also assess whether a stable non-perturbative denition can be provided via a random matrix model. The higher topology contributions toN = 1 JT supergravity observables were systematically computed in [146], where the topological expansion was matched to certain random matrix models via the loop equations. Non-perturbative stable denitions were then given in [7, 179], using particular matrix models and their orthogonal polynomial formulation. TheN = 1 JT supergravity theory was rst studied in [180, 181], where a space-time formulation was provided using two-dimensional superspace [182]. After integrating over the fermionic variables and solving the equations of motion of the auxiliary elds, one is left with the bosonic elds (g ;) and their supersymmetric partners ( ; ), the gravitino and dilatino (here is a spinor index). For the explicit action see for example equation (2.3) in [183]. Moreover, as shown in [184], JT supergravity is a topological theory, with no bulk degrees of freedom, as can be seen by rewriting the two-dimensional action as a OSp(1j2) BF theory (see for example [185]). Altogether, this means we should only worry about the degrees of freedom at the boundary of the manifold. Performing the same computation that lead to the Schwarzian quantum mechanics (7.31) in ordinary JT gravity, one nds the boundary degrees of freedom are controlled by the following supersymmetric quantum mechanics [186] I N =1 Sch [(u);(u)] = 1 2 Z 0 du Sch(tan(=2);u)(1 0 ) + 000 + 3 0 00 ; (7.50) where (u) is a Grassmann eld, the supersymmetric partner of (u). The boundary conditions satised in each case are given by (u +) =(u) + 2 ; (u +) =(u) : (7.51) This supersymmetric quantum mechanic theory is called the super Schwarzian and was rst intro- duced in [187] from theN = 1 SYK. Setting (u) = 0 and using (7.29), we recover the bosonic Schwarzian action (7.31). The upshot is that theN = 1 JT supergravity partition function with disc topology is given by the path integral of this theory Z N =1 0 () =e S 0 Z DDe I N=1 Sch [(u);(u)] ; (7.52) where the factor e S 0 comes from the Euler charateristic. The boundary conditions of the gravi- tational path integral are analogous to the bosonic conditions in (7.16), but are more naturally 163 formulated in superspace (see [186] for details). 11 The disc path integral (7.52) was evaluated exactly in [171] by (similarly as in the JT gravity) showing it is one-loop exact. To evaluate the one-loop contribution we write (u) as in (7.32) and expand (u) in a Fourier series (u) = X m2(Z+ 1 2 ) m e iu 2m ; (7.53) where m = m is a Grassmann number and m2 (Z + 1 2 ) in order to satisfy the anti-periodic boundary condition (7.51). Inserting in the super Schwarzian action (7.50) and expanding to quadratic order we nd I N =1 Sch [(u);(u)] =I Sch [(u)] O(" 2 n ) 1 2 Z 0 du " 2 0 00 1 2 2 2 0 # +::: ; =I Sch [(u)] O(" 2 n ) 1 2 2 3 X m2(Z+ 1 2 ) m(1 4m 2 )Re( m )Im( m ) +::: ; (7.54) where the rst term is the expansion of the Schwarzian action (7.33). From this expression we see that apart from the three bosonic zero modes n = 0;1 corresponding to SL(2;R), there are two additional fermionic zero modes at m =1=2. Altogether, these corresponds to OSp(1j2) transformations that leave the super Schwarzian action (7.50) invariant. 12 To get a nite result for the path integral, we must not integrate over these zero modes. The measureDD in (7.52) is again computed by noting the path integral is over a symplectic manifold, in this case corresponding to the super Virasoro group, moded out by OSp(1j2). The symplectic form to quadratic order in the elds is given in equation (C.7) of [146] = Bosonic + 1 2 Z 0 du 2 2 d(u)^d(u) 4d 0 (u)^d 0 (u) +::: = Bosonic + 2 2 X m 3 2 (1 4m 2 ) dRe( m )^ dRe( m ) + dIm( m )^ dIm( m ) +::: (7.55) where Bosonic is already given in (7.34). Using this to compute the measure associated to (u) we 11 In particular, we require the dilation to vanish at the boundary. Same as in the bosonic case, we are always implicitly taking the limit ! 0 meaning the boundary @M is asymptotically close to the boundary of the Poincare disc (see gure 7.1). 12 For some details on OSp(1j2) see for example section 3.5.1 in [146]. 164 can put all together in (7.52) and solve the additional Grassmann integrals 13 Z N =1 0 () =e S 0 e 2 = Y n2 2 n Y m 3 2 m 2 =e S 0 r 2 e 2 = ; (7.56) where in the second equality we regulated the innite product similarly as before, using the Hurwitz zeta function instead. This is the exact result for theN = 1 JT supergravity disc partition function. Dening the spectral density as in (7.38) we nd that in this case it is given by N =1 0 (E) =e S 0 p 2 cosh(2 p E) p E : (7.57) 7.2.1 Topological expansion and random matrices Having computed the disc partition function, let us now consider the full topological expansion of the Euclidean partition function. Similarly as before, the integral over the dilaton eld (x ) xes the path integral over metrics with constant negative curvatureR =2. The Euler characteristic controls the contribution from each topology, so that the general expansion takes the same form as in (7.39). The dierence is that in this case the integral is over super Riemann surfaces. To mimic the decomposition of the higher genus contributions shown in gure 7.2, we need to compute the N = 1 trumpet partition function which generalizes (7.41). An analogous computation to the one leading to (7.56) gives Z N =1 Trumpet (;b) = 1 p 2 e b 2 =4 ; (7.58) see appendix C of [146] for details. The contribution to the topological expansion at any order in genus g is constructed using the trumpet as in (7.44). The volumes of moduli spaces of Riemann surfaces V g (b 1 ;:::;b n ) are now replaced by the corresponding volumes associated to super Riemann surfaces b V g (b 1 ;:::;b n ). These super volumes where dened and carefully studied in [146]. A simple way of thinking about them comes from the realization that the moduli space of super Riemann surfaces can be described from the moduli space of ordinary (bosonic) Riemann surfaces with a spin connection (see appendix A of [146]). Broadly speaking, all the spin connection does is determine the behavior of the fermion elds in the supergravity theory as they are transported across a closed loop in the manifold, either 13 The analog of the determinant of that appropriatelly deals with both the bosonic and fermionic variables is the Berezinian. For example see the discussion around equation (3.67) in [146]. An additional factor of (2) 1 was included in for convenience. 165 Figure 7.3: Spin structures on the circle and torus, where in each case we indicate whether it is even or odd. periodic (Ramond) or anti-periodic (Neveu-Schwarz). For instance, on the asymptotic boundary the Grassmann eld (u) of the super Schwarzian quantum mechanics satises Neveu-Schwarz boundary conditions (7.51). 14 Higher topology surfaces allow for more interesting spin structures. It is instructive to consider the example of a torus, which can be described by the (x 1 ;x 2 ) plane with the identications x i x i + 2. Since there are two closed loops in the manifold, there are four dierent spin structures one can dene, given by (x 1 + 2;x 2 ) = (x 1 ;x 2 ) ; (x 1 ;x 2 + 2) = (x 1 ;x 2 ) ; (7.59) where (x 1 ;x 2 ) is a fermion eld. In the right diagram of gure 7.3 we indicate the four possible combinations. Spin structures are readily classied in terms of their parity (1) , where corre- sponds to the number of chiral zero modes for a given spin structure. For instance, a fermion in S 1 has a single component and its Dirac equation is given by@ x 1 = 0, wherex 1 x 1 +2 parametrizes the circle. While for the Neveu-Schwarz spin structure there is no zero mode (1) = 1, for Ramond there is one zero mode, meaning (1) =1 (see left diagram in gure 7.3). As explained in [146] there are two distinct ways of dening the super volumes b V g;n (b 1 ;:::;b n ), depending on how dierent contributions from the spin structure are incorporated. One can either consider the sum or dierence between even and odd spin structures, where the later corresponds to inserting a factor of (1) in the path integral. Schematically, the volumes dened in each case are given by b V g;n (b 1 ;:::;b n ) = (even) (odd) : (7.60) 14 Similarly, the trumpet partition function in (7.58) also has anti-periodic conditions for the fermions, not only at the asymptotic boundary, but also in the interior and geodesic boundary b. 166 The volumes in each case turn out being radically dierent: while b V g;n (b 1 ;:::;b n ) are non-trivial and can be computed from a closed recursion relation given in equation (5.42) of [146], all the volumes b V + g;n (b 1 ;:::;b n ) vanish identically. This denes two very distinct theories ofN = 1 JT quantum supergravity. The topological expansion in each case is give by Z ( 1 ;:::; n ) = 1 X g=0 e S 0 2(g1)+n Z g ( 1 ;:::; n ) ; (7.61) where the contribution for any given genus is obtained from the analogous expression to (7.44) corresponding to the decomposition shown in gure 7.2 15 Z g ( 1 ;:::; n ) = n Y i=1 Z 1 0 db i b i Z N =1 Trumpet ( i ;b i ) b V g;n (b 1 ;:::;b n ) : (7.62) Using the corresponding super volumes, we can explicitly compute the partition function to arbi- trary orders. The result is particularly spectacular for the supergravity which sums over all spin structures, considers the sum over even and odd spin structures, since using b V + g;n = 0 we nd Z + ()'e S 0 r 2 e 2 = ; Z + ( 1 ; 2 )' 2 p 1 2 1 + 2 ; (7.63) together with Z + ( 1 ;:::; n>2 )' 0, where we have used the volume for g = 0 and n = 2 does not vanish (see footnote 15). The symbol' reminds us that although these results hold to all order in a perturbative expansion in e S 0 , in principle there could be non-perturbative corrections. Similarly as in the JT gravity case, the next question we want to address is whether there is a random matrix model we can construct which reproduces the perturbative expansion in (7.61). The vanishing of the perturbative expansion for theN = 1 theory dened through the volumes b V g;n (b 1 ;:::;b n ) chimes with the cancellation of the loop equations of a Hermitian matrix model whose leading eigenvalue spectral density is supported on the whole real line, studied in subsection 6.1.3. As rst noted in [146], the perturbative expansion is reproduced by an ensemble of random Hermitian matrices Q2C NN in the following way Z + ( 1 ;:::; n )' n Y i=1 p 2 Tre i Q 2 c (7.64) 15 Forg = 0 andn = 2 the corresponding supervolume is formally dened as b V 0;2 (b1;b2) = 2(b1b2)=b1. Comparing with the bosonic case in footnote 8, there is an additional factor of two which comes from summing over the spin structures of the double trumpet geometry that contributes to Z 0 (1;2). Although both boundaries 1 and 2 have Neveu-Schwarz boundary conditions for the fermions, when gluing the two trumpets there is a freedom when identifying the fermions in each trumpet left = right . Irrespectively of whether we have an insertion of (1) or not, both cases contribute in the same way (see section 2.4.3 in [146]). 167 where the leading genus eigenvalue density is given by 0 (q) = cosh(2q) ; (7.65) with q 2 R. Using this together with (6.31) and (6.42) coming from the loop equations it is straightforward to match with the gravity results in (7.64). For the other supergravity theory a matrix model description of the topological expansion was given in [146], using a dierent kind of matrix ensemble to the ones introduced here. 7.2.2 Non-perturbative denition We can now investigate whether the random matrix model can supply a stable non-perturbative def- inition of the JT quantum supergravity theory. From here onwards we focus on the Z + ( 1 ;:::; n ) theory, the other case being studied in [179, 188]. While directly computing non-perturbative corrections of orderO(e e S 0 ) to the supergravity partition functions in (7.63) is not tractable, assuming (7.64) holds beyond perturbation theory, we can attempt to use the matrix model to compute non-perturbative contributions to the path integral. This is the perfect setup to study these eects, given that for Z + ( 1 ;:::; n>2 ) there are only non-perturbative contributions. To study non-perturbative eects we use the method of orthogonal polynomials, discussed in section 6.2. In particular, in subsection 6.2.2 we carefully studied the double scaling of the critical potentials in (6.75), whose leading eigenvalue density (6.74) behaves like 0 () 2k . Expanding the leading eigenvalue density required to describe JT supergravity (7.65) 0 (q) = cosh(2q) = 1 1 X k=0 (2) 2k (2k)! q 2k = 1 + 2q 2 + 2 3 3 q 4 +O(q 6 ) ; (7.66) we note it can be built as a superposition of precisely these kind of models. In particular, we showed that the double scaling limit is controlled by the functionr(x) satisfying the string equation (6.99), that we rewrite here for convenience 1 X k=1 t 2k K 2k +r(x)x = 0 ; (7.67) whereK 2k are dierential polynomials inr(x) computed from the recursion relation in (6.95). The coecients t 2k control the superposition of the critical potentials according to (6.97), essentially controlling the term q 2k in the leading spectral density. By tuning the coecients t 2k so that we 168 reconstruct 0 (q), we should be able to assemble the right double scaled model that reproduces the perturbative gravitational answers in (7.63). Moreover, since all the critical potentials used to build the matrix model (6.75) are stable, we do not expect to nd non-perturbative issues in the denition of the matrix model, like we did for the bosonic JT gravity case (7.49). Assuming (7.64) holds beyond perturbation theory, the partition function can be computed from the expectation value of the insertion of the matrix operator Tre i Q 2 , as written in equations (6.107) and (6.109). For example, for the single boundary case we have 16 Z + () = p 2 Tre Q 2 = p 2 X s= Z 1 dxhxje Hs jxi ; (7.68) whereH s =(~@ x ) 2 +[r(x) 2 s~r 0 (x)]. This provides an extremely concrete formalism to compute the full gravitational partition function ofN = 1 JT supergravity, including both perturbative and non-perturbative contributions in~ =e S 0 . The data that species dierent doubled scaled matrix model is given by the following parameters Matrix model data : ;ft 2k g k2N : (7.69) Perturbation theory Our rst task is to use perturbation theory in~ =e S 0 to x the parameters (7.69) of the double scaled model that ensures the matching in (7.64) with the supergravity results (7.63). Writing a r(x) in a perturbative expansion r(x) =r 0 (x) + 1 X n=1 r n (x)~ n ; (7.70) inserting in the string equation (7.67) and using K 2k =r(x) 2k+1 +O(~ 2 ) (6.94), we nd r 0 (x) " 1 X k=1 t 2k r 0 (x) 2k+1 +x # = 0 : (7.71) For any given value of x, there are two possible solutions corresponding to the vanishing of each factor. To obtain a non-trivial, single valued and continuous function r 0 (x) forx2R we must take r 0 (x) : 8 > > > < > > > : 1 X k=1 t 2k r 0 (x) 2k +x = 0 ; x 0 ; r 0 (x) = 0 ; x 0 : (7.72) 16 The factor of 1= 2 appearing in (6.107) has been absorbed into the denition of the matrix Q. 169 The detailed behavior of this function is determined by the coecients t 2k , which can be xed by matching to the leading~ behavior of the single boundary partition function in (7.68). T do so, let us dene the usual Hermitian momentum operator ^ p =i~@ x with eigenstates ^ pjpi =pjpi. Usingfjpig p2R forms a complete basis, we can derive the following useful identity hx 1 je Hs jx 2 i = 1 p 2~ Z +1 1 dphx 1 je [^ p 2 +r 0 (x) 2 ] jpie ipx 2 =~ +O(~) = 1 2~ e r 0 (x 1 ) 2 Z +1 1 dpe p 2 e ip(x 2 x 1 )=~ +O(~) = 1 2~ p exp r 0 (x 1 ) 2 1 x 2 x 1 2~ 2 +O(~) (7.73) where we have usedhxjpi = e ixp=~ = p 2~ and [^ x; ^ p] = i~. Note that to leading order the result is independent ofs =. Using this together with the solution for r 0 (x) (7.72) we can easily evaluate the right-hand side of (7.68) p 2 Tre Q 2 = 2 p 2 2~ p Z 1 dxe r 0 (x) 2 +O(~) = 1 ~ r 2 " + 1 X k=1 t 2k 2k Z +1 0 dr 0 r 2k1 0 e r 2 0 # +O(~) = 1 ~ r 2 " + 1 X k=1 t 2k k! k # +O(~) ; (7.74) where in the second equality we have changed the integration variable to r 0 and computed the Jacobian using (7.72). We have also used the boundary conditions of the string equation (6.96) imply r 0 (1) = +1. Comparing this expression to the gravity result in (7.63) we see the matrix model answer has precisely the right structure. Agreement between the two expression completely xes (7.69) according to (;t 2k ) = 1; 2k k! 2 (7.75) This gives an unambiguously an non-perturbative denition of the double scaled model through the string equation (7.67). With this choice, the leading spectral density 0 (q) takes the form given in (7.66). The leading behavior ofr 0 (x) (7.72) in the regionx 0 can be resummed and written in terms of a modied Bessel function, I 0 (2r 0 ) 1 +x = 0. From this it is straightforward to numerically solve this constraint and plot r 0 (x) in gure 7.4. While for negative x there are two branches 170 Figure 7.4: Plot of the leading solution r 0 (x) obtained from (7.72) after xing t 2k according to (7.75). In the region x < 0 there are two solutions to the implicit constraint, indicated in red (solid) and blue (dashed). We chose the red branch, which satises the boundary condition (6.96) required when solving the string equation . that satisfy the implicit constraint only the positive branch satises the boundary conditions of the string equation (6.96). Note the qualitative behavior of r 0 (x) at x 0 is an agreement with the solution to the discrete string equation for the k = 1 case displayed in gure 6.3. Fixing the parameters (;t 2k ) as in (7.75) ensures the matching to Z + () in (7.63) only to leading order. Let us now compute the higher ~ corrections in (7.74), which according to the supergravity result should vanish to all orders. To do so, we must rst obtain the subleading terms r n (x) with n 1 in (7.70) by perturbatively solving the string equation (7.67). There are two dierent perturbative expansions, centered around positive or negative large x. It turns out the relevant expansion for comparing with supergravity corresponds to the one centered around x positive. Moreover, to recursively solve the string equation (7.67) it is convenient to introduce an additional parameter c2R in the following way 1 X k=1 t 2k K 2k +r(x)x =~c ; (7.76) so that c = 0 corresponds to the case of interest. Inserting (7.70) in this dierential equation it is straightforward to solve for the rst few orders and nd r(x) =r 0 (x) + ~c x 1 +t 2 ~ 2 (1c 2 ) x 3 + ~ 4 (1c 2 ) x 6 t 2 2 (10 3c 2 )t 4 (4c 2 )x +O(~ 6 ) : (7.77) Note the expansion has a nite radius of convergence, given that all corrections diverge as we approach x = 0. Moreover, the whole perturbative series vanishes when c = 0, that corresponds to the case we are ultimately interested in. This is no accident and follows from the observation 171 that r(x) = 0 is an exact solution to the full string equation when c = 0. This means the function r(x) receives no perturbative corrections and is given by r 0 (x) plotted in gure 7.4 to all orders in perturbation theory. Let us now use this to explicitly compute the corrections to the right hand side of (7.68). To do so, we use the following expansion of the resolvent of the operatorH s worked out by Gel'fand and Dikii [189] hxj 1 H s jxi = 1 ~ 1 X p=0 1 () p+1=2 (2p 1)!R p [u s (x)] (4) p p!(p 1)! ; (7.78) where< 0 and we have denedu s (x) =r(x) 2 s~r 0 (x). The Gel'fand-Dikii functionalsR p [u s (x)] are polynomials in u s (x) and its derivatives computed from the following recursion relation R p+1 = 2p + 2 2p + 1 u s (x)R k ~ 2 4 R 00 k 1 2 Z x d xu 0 s ( x)R k ; (7.79) with R 0 = 1. Comparing with the expressions in [189] there are some dierent factors in the resolvent and recursion relation since we are using a dierent normalization which ensures R p [u s (x)] =u s (x) p +O(~ 2 ). Applying an inverse Laplace transformation in to this expansion for the resolvent we nd hxje Hs jxi = 1 2~ p 1 X p=0 () p p! R p [u s (x)] ; (7.80) which is more convenient for evaluating higher ~ corrections tohTre Q 2 i in (7.68). While this asymptotic series formula is valid to all orders in~, we cannot exchange the innite series with the x integral in (7.68). Instead we must solve the series order by order in~ and only then compute the integral. We can do this by using the recursion relation (7.79) and work out their general expansion for the rst few orders in~ R p =u p s ~ 2 12 p(p 1)u p3 2u s u 00 s + (p 2)(u 0 s ) 2 + ~ 4 1440 p(p 1)(p 2)u k6 s h 24u 3 s u (4) s + + 48(p 3)u 2 s u 0 s u 000 s + 36(p 3)u 2 s (u 00 s ) 2 + 44(p 3)(p 4)u s (u 0 s ) 2 u 00 s + 5(p 3)(p 4)(p 5)(u 0 s ) 4 i +::: (7.81) Inserting this into (7.80) and solving the series order by order in~ we nd hxje Hs jxi = e us 2~ p 1 ~ 2 2 12 2u 00 s (u 0 s ) 2 ~ 4 3 1440 24u (4) s 48u 0 s u (3) s 36(u 00 s ) 2 + 44 2 (u 0 s ) 2 u 00 s 5 3 (u 0 s ) 4 +O(~ 6 ) : (7.82) 172 This expression does not include the the~ dependence inu s (x) =r(x) 2 s~r 0 (x), both explicitly and implicitly through the r(x) expansion in (7.70). Taking this into account we obtain the expansion written directly in terms of r n (x) hxje Hs jxi = e r 2 0 2~ p 1 +~(sr 0 0 2r 0 r 1 ) + ~ 2 12 2 [(r 2 0 ) 0 ] 2 12 r 2 1 + 2r 0 r 2 sr 0 1 + + 2 12(r 0 r 1 ) 2 + (r 0 0 ) 2 2r 0 r 00 0 12sr 0 r 1 r 0 0 +O(~ 3 ) : (7.83) To construct the expectation value of Tre Q 2 we should in principle integrate this in the range x2 (1;], as indicated in (7.68). The rst term in this expression gives the leading contribution we have already computed in (7.74). For the subleading terms we have to be more careful with the integration region, since (as previously mentioned) the ~ expansion is centered around x = +1 which has a nite radius of convergence all the way up to x = 0. Therefore, all subleading terms should only be integrated in the range x2 [0;]. Moreover, since r(x) vanishes for x positive to all orders in perturbation theory when c = 0 (7.77), we conclude all the subleading contributions tohTre Q 2 i vanish and we are only left with p 2 Tre Q 2 ' 1 ~ r 2 e 2 = ; (7.84) where the symbol' indicates the equality holds to all orders in perturbation theory. Altogether, this is in precise agreement to the supergravity result in (7.63). For the expectation value of several insertions of Tre Q 2 we can use the formulas in (6.109). For instance, when we have two insertions we can easily evaluate the leading genus contribution using (7.73) 2 Y i=1 p 2 Tre i Q 2 c = 1 ~ 2 p 1 2 Z 1 dx 1 Z +1 dx 2 e 1 r 0 (x 1 ) 2 r 0 (x 2 ) 2 1 + 2 1 2 ( x 2 x 1 2~ ) 2 +O(1) : (7.85) Changing the integration variables to x 1 =~(w 1 w 2 ) + and x 2 =~(w 1 +w 2 ) + the integrals decouple after we expand the exponentials e i r 0 (x i ) 2 ' e i r 0 () 2 to leading order in ~. The remaining (w 1 ;w 2 ) integrals can be easily solved and we nd 2 Y i=1 p 2 Tre i Q 2 c = 2 p 1 2 1 + 2 e ( 1 + 2 )r 0 () 2 +O(1) : (7.86) Using r 0 () = r 0 (1) = 0, the leading behavior precisely matches with the supergravity result in (7.63). Although this same procedure can be applied to compute the leading genus contribution 173 for three and more insertions of Tre Q 2 (see appendix A in [7]), it becomes increasingly tedious. Instead we can use a compact formula derived in [190, 191] for the leading behavior of double scaled Hermitian matrix models with an edge critical potential (see table 6.1). While those models have several dierences compared to the one studied here, observables are also computed from an almost identical eective quantum mechanical system, with the dierence there is no sum over spin s = as in (7.68). Taking this into account, we can apply the general formula of [152, 190, 191] to this case and nd n Y i=1 p 2 Tre i Q 2 c = 2 n 2 +1 p 1 n 2 n=2 T h (~@ x ) n2 e T r 0 (x) 2 i x= ; (7.87) where we have dened T = P n i=1 i . For n = 2; 3 we recover the results that can be explicitly computed using the procedure applied in (7.85). Applying this formula for n 3 we nd the leading contribution always vanishes given that r 0 (x) and all its derivatives vanish at x = > 0. This matches with the result from the supergravity computation. In summary, we have shown how partition function of JT supergravity with a sum over spin structures Z + ( 1 ;:::; n ) is equivalent to a double scaled model constructed from a particular superposition (7.75) of interior critical potentials (6.97). The advantage of this formulation when compared to the one using the matrix model loop equations, is that it allows for non-perturbative computation of observables. Non-perturbative physics Let us now turn our attention to the exact computation of Z + () using the non-perturbative completion given by the matrix model according to (7.68). Instead of computinghTre Q 2 i it is convenient to calculate the expectation value of the eigenvalue spectral density of the matrix model (q) = Tr(Qq), which using (6.115) can be written in terms of the self-reproducing kernel h(q)i =jqj X s= L s (q 2 ;q 2 ) ; L s (E 1 ;E 2 ) = Z 1 dx E 1 ;s (x) E 2 ;s (x) ; (7.88) where E;s (x) are eigenfunctions ofH s with eigenvaluesE (6.111). The expectation value of two insertions of (q) can also be expressed in terms of the kernel L s , which as show in (6.113) can be exactly integrated. 174 The eigenvalue spectral density of the matrix model (q) should not be confused with the supergravity energy spectral density grav (E), dened from Z + () = Z +1 0 dE grav (E)e E : (7.89) Using (7.64) they can be easily related to each other grav (E) = r 2 E ( p E) : (7.90) To leading order, this relation is trivially satised by (7.57) and (7.65). A toy model: Before considering the full model, it is instructive to work out things for a simple toy model that is relevant to JT supergravity and where everything can be computed analytically. The model is dened in the following way 17 r(x) = ~c x ; =) H s =(~@ x ) 2 + ~ 2 c(c +s) x 2 ; (7.91) with x > 0. When c = 0;1 the function r(x) is an exact solution to the string equation (7.76). Note the operatorH s is nothing more than a one dimension Hydrogen atom. Its eigenfunctions E;s (x) (6.111) can be easily computed and written in terms of a Bessel function E;s (x) = N ~ r x 2 J s p Ex=~ ; s =c + s 2 ; (7.92) where the other independent solution is discarded since it is not regular at the origin. To write the solution in this way we have also used that s 2 = 1. We are still left with a normalization constantN that is left arbitrary since the state is not normalizable. This can be potentially problematic, given that theq dependence of observables such ash(q)i in (7.88) depends on the value ofjNj 2 . The way we can unambiguously x this constant is by taking the classical limit~! 0 ofh(q)i using a similar approach as in (7.73) h(q)i =jqj X s= Z 1 dxhxj(H s q 2 )jxi =jqj X s= Z 1 dx Z +1 1 dpjhxjpij 2 hxj(p 2 +r 0 (x) 2 q 2 )jxi +O(~) =jqj X s= Z 1 dx [q 2 r 0 (x) 2 ] 2~ p q 2 r 0 (x) 2 +O(~) ; (7.93) 17 This model is very much related to the Bessel model extensively studied for example in [146, 177, 179]. 175 where we have inserted a complete set of momentum eigenstates and solved the additional integral. The key dierence is that we have written the expectation value in terms of the norm of momentum stateshxjpi which have an unambiguous normalization. Comparing this with the expression fro h(q)i in terms of E;s (x) (7.88) we nd lim ~!0 j E;s (x)j 2 = 1 2~ p E + oscillating ; x> 0 ; (7.94) where we have used r 0 (x) always vanishes for x > 0. Since E;s (x) for large positive x behaves like a free particle, we expect to have additional oscillating terms which average to zero. We can use this condition to x the normalization of the eigenfunctions E;s (x). For this toy model, the classical limit (7.92) gives lim ~!0 j E;s (x)j 2 = jNj 2 2~ p E + jNj 2 2~ p E sin 2 p Ex ~ s ; (7.95) which comparing with (7.94) xesjNj 2 = 1. Having xed the normalization of the eigenfunctions, we can now easily compute the self repro- ducing kernel, using the integrated form in (6.113) L s (E 1 ;E 2 ) = 1 2 ~ 2 J s ( 1 ) 2 J s1 ( 2 ) 1 J s1 ( 1 )J s ( 2 ) 2 1 2 2 ; L s (E;E) = 2~ 2 J s () 2 J s1 ()J s+1 () ; (7.96) where i = p E i =~. For the case that is relevant to JT supergravity (c;) = (0; 1), the kernel further simplies to L s (E 1 ;E 2 ) (c;)=(0;1) = 1 ~ 2 p 1 2 sin( 1 + s ) 2 cos( 2 + s ) 1 cos( 1 + s ) sin( 2 + s ) 2 1 2 2 ; L s (E;E) (c;)=(0;1) = 1 2~ p E s 4E sin 2 p E ~ : (7.97) where we have dened s = 4 (2 s 1). The expectation value of any matrix model observable can be computed from these expressions, including both perturbative and non-perturbative contri- butions in~. For example, the expectation value of one or two insertions of the spectral density 176 Figure 7.5: Plot of the spectral density of the toy model with (c;) = (0; 1) in (7.98), that is rele- vant for JT supergravity. While the constant black curve (dashed) corresponds to the perturbative result, in blue we include also non-perturbative eects, which are extremely relevant for small q. operator (q) give (6.115) h(q)i = X s= 1 2~ s 4q sin(2q=~) ; h(q 1 )(q 2 )i c =h(q 1 )ijq 1 j(q 2 1 q 2 2 ) 1 2 2 q 2 1 +q 2 2 (q 2 1 q 2 2 ) 2 + + (q 1 +q 2 ) 2 cos [2(q 1 q 2 )=~] + (q 1 q 2 ) 2 cos [2(q 1 +q 2 )=~] 2(q 2 1 q 2 2 ) 2 : (7.98) The observables have a nite number of perturbative terms in ~, as well as non-perturbative corrections, given by fast oscillating functions. The case of a single eigenvalue spectral density is quite remarkable, given that when summing over the spin s = degrees of freedom, the non- perturbative contribution exactly cancel and we are left with the exact resulth(q)i = 1=~ (see gure 7.5). Moreover, note thath(q)i = 1=~ corresponds precisely to the low q behavior of the spectral density 0 (q) in (7.66) required to describe JT supergravity. While non-perturbative contributions coming from each spin s = sector cancel out for the expectation value of single trace observables, we see from (7.98) this is not the case for higher trace observables. JT supergravity: Let us now consider the full double scaled that is relevant for describing JT supergravity, dened from the values of and t 2k indicated in (7.75). The analytic computation of non-perturbative contributions is much more challenging in this case, as it is not possible to write a simple solution E;s (x) as in the toy model (7.92). However, we can make some progress 177 by computing the eigenfunction E;s (x) in the usual WKB approximation lim ~!0 E;s (x) = A + exp h i ~ R x x min d x p Er 0 ( x) 2 i +A exp h i ~ R x x min d x p Er 0 ( x) 2 i (Er 0 (x) 2 ) 1=4 ; (7.99) where the parameterx min is dened fromr 0 (x min ) 2 =E, so that this is the solution in the classically allowed region. The challenge is xing the undetermined constants A . To do so, we note that for any double scaled model at suciently large x the potential r(x) approaches r(x) =~c=x of the toy model (7.91). Therefore, for large x the general WKB approximation in (7.99) should match to the classical limit of the eigenfunctions of the toy model (7.92), given by lim ~!0 E;s (x) = 1 p ~ p E cos " p Ex ~ 4 (1 + 2 s ) # : (7.100) Comparing with (7.99) we obtain the WKB approximation for an arbitrary model lim ~!0 E;s (x) = cos h 1 ~ R x x min d x p Er 0 ( x) 2 4 (1 + 2 s ) i p ~(Er 0 (x) 2 ) 1=4 : (7.101) Using this expression we can compute the self-reproducing kernel L s (E 1 ;E 2 ) and therefore the expectation of any observable to leading order. The advantage of this approach is that this method captures not only the leading perturbative but also the leading non-perturbative contribution. To show this, let us compute the diagonal components of the kernel using the integrated formula in (6.114), which gives lim ~!0 L s (E;E) = 1 2~ Z x min dx p Er 0 (x) 2 s 4E sin 2 ~ Z x min dx p Er 0 (x) 2 c ; (7.102) where we have used > 0, r 0 (x) = 0 for x> 0 and written s =c +s=2 explicitly. From this we can easily compute the eigenvalue spectral densityh(q)i (7.88), which gives lim ~!0 h(q)i = 1 ~ 0 (q) X s= s 4jqj sin " 2 ~ Z jqj d q 0 ( q)c # ; (7.103) where we used 0 (q) = jqj 2 X s= Z x min dx p q 2 r 0 (x) 2 ; Z jqj d q 0 ( q) = 1 Z x min dx p q 2 r 0 (x) 2 ; (7.104) with 0 (q) the leading perturbative contribution in (7.93) (given by (7.66) for JT supergravity theory). 178 -25 -20 -15 -10 -5 0 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -50 -40 -30 -20 -10 0 10 20 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Figure 7.6: The solid blue curve corresponds to the full non-perturbative numerical solution r(x) to the string equation (7.106) in the k max =6 truncation with ~=1. The dashed red curve is the leading genus solution r 0 (x) in (7.72), which is actually the exact solution to all orders in perturbation theory. The second term in (7.103) is the interesting one, which gives the leading non-perturbative contribution toh(q)i. 18 The peculiarity is that the contributions froms = precisely cancel each other, so that there is eectively no leading non-perturbative eect. There can still be subleading non-perturbative contributions, schematically given by h(q)i = 1 ~ 0 (q) +~ n sin(#=~) +::: ; (7.105) with n > 0. For the toy model (7.91) we have seen in (7.98) that all non-perturbative terms in h(q)i vanish. Is this also the case for the matrix model that describes JT supergravity? To answer this, we need to compute the full eigenvalue spectral densityh(q)i, without making any approximation. Since there is now way of performing the calculation analytically, we proceed numerically. The rst step is to solve the string equation (7.67) for r(x). To make sense of it as a nite order dierential equation, we follow [188] and introduce a truncation by only including contributions from t 2k up to some maximum value k max 1 X k=1 t 2k K 2k +r(x)x = 0 ! kmax X k=1 t 2k K 2k +r(x)x = 0 ; (7.106) where (;t 2k ) are given in (7.75). For high enough values ofk max any artifacts due to the truncation are, at low enough energies, indistinguishable from other numerical errors due to discretization to 18 A dierent calculation in appendix E of [146] gives similar (and in some sense equivalent) result for other random matrix ensembles (see also [188]). 179 -0.6 -0.4 -0.2 0 0.2 0.4 0 1 2 3 4 5 6 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0 1 2 3 4 5 6 Figure 7.7: In the left diagram we plot each spin contribution s = (7.108) to the spectral density, with the dashed line corresponding to the perturbative answer. In the right diagram we sum both contributions and compare it to the perturbative answer 0 (q) in (7.66) (dashed line). ndr(x) or when subsequently solving the spectral problem numerically. From the general structure ofK 2k indicated in (6.94) we see the dierential equation for r(x) is of order 2k max . The boundary conditions at large values ofjxj are xed by the leading~ solution r 0 (x) in (7.72) lim jxj!1 @ n r(x) @x = lim jxj!1 @ n r 0 (x) @x ; n = 0; 1;:::;k max 1 : (7.107) Similar truncation procedures have been successfully applied to other double scaled matrix models in relation to JT gravity and supergravity [9, 192]. In the left diagram of gure 7.6, the blue solid line corresponds to the numerical solution forr(x) with~ = 1 and a truncation with k max = 6. Working with higher truncations does not generate any substantial dierence, at least for the numerical precision required for the computations here. The dashed line corresponds to r 0 (x) in (7.72), that is actually the exact solution to all orders in perturbation theory. The substantial dierence between r(x) and r 0 (x) for small values of x is entirely due to non-perturbative eects. On the right diagram of that same gure we plot r(x) 2 s~r 0 (x), that is the potential appearing in the operatorH s (7.68). Using these solutions we can build the operatorH s and compute the corresponding eigenfunc- tions E;s (x), similarly as in [9, 177, 179, 188]. Although the eigenfunctions are not normalizable, the normalization constant is xed from the condition in (7.94). Using (7.88) this allows us to com- pute the full spectral densityh(q)i, including all perturbative and non-perturbative contributions in~. The nal result is shown in gure 7.7, where in the left diagram we plot each of the s = 180 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 0 2 4 6 8 10 12 14 16 18 20 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0 1 2 3 4 5 6 Figure 7.8: In the left diagram we plot each spin contributions = to the spectral density for = 10. In this regime, we observe good matching with the WKB expression in (7.103) corresponding to the gray curves. In the right diagram we sum both contributions and obtain the expected matching to the perturbative result 0 (q) with = 10. contributions h s (q)i =jqjL s (q 2 ;q 2 ) ; (7.108) the dashed line in that diagram corresponding to the leading perturbative result 0 (q)=2. Summing these functions, we obtain the full spectral densityh(q)i in the right diagram. Comparing with the perturbative answer we observe there are indeed subleading non-perturbative corrections of the schematic form given in (7.105). Since the dierence between the perturbative and full spectral densities in the right diagram of gure 7.7 is not particularly big, we want to make sure it is not generated by numerical uncertainties but real physics. One way of checking this is by taking smaller values of ~ so that subleading contributions (7.105) are further suppressed and the non-perturbative eects are entirely controlled by the WKB approximation in (7.103). As noticed in [188], instead of decreasing the value of ~, it is numerically more convenient to increase . This has the same eect, as larger values of means the x integral used to compute the kernel L s (E 1 ;E 2 ) in (7.88) gets larger contribution from the classical region of the potential inH s , suppressing the quantum eects. In the left diagram of gure 7.8 we plot each spin contribution s = to the spectral density when = 10 (instead of = 1 (7.75)). The gray curves in that diagram correspond to the WKB approximation in (7.103), where we nd good agreement. 19 In the right diagram we sum both 19 A similar matching for a dierent class of double scaled matrix models that are relevant for other JT supergravity 181 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 q -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 Figure 7.9: In these diagrams we have isolated the non-perturbative contributions to the spectral density for = 1 and = 10. For large we observe the non-perturbative eects are supressed, as predicted by the WKB approximation in (7.103). contributions and plot the full spectral density. Comparing with the right diagram of gure 7.7 we observe the full solution for = 10 is much closer to the perturbative answer when = 1. This provides good evidence that points towards the non-perturbative eects for = 1 being real physics instead of numerical error. To see this more clearly, we can isolate the non-perturbative contributions in each case by substracting the perturbative piece 0 (q). In gure 7.9 we plot this for = 1 and = 10. As expected, we observe the non-perturbative corrections that are present for = 1 are supressed as increases and the answer is dominated by the WKB approximation (7.103). Altogether we have shown how the Hermitian matrix model studied in subsection 6.2.2 can be used to provide a stable non-perturbative denition. Moreover, non-perturbative eects can be computed explicitly, using a combination of analytic and numerical techniques. Similar non- perturbative denitions for other JT gravity and supergravity theories has been explored in the literature [7, 9, 177, 179, 188, 192]. theories was performed in [188]. 182 Appendix A Appendix A.1 Symmetry constraints on the vacuum stress tensor In this appendix we analyze how the vacuum stress tensorhT i 0 is constrained by symmetry conditions of the manifold where the QFT is dened. Let us start by considering the simpler case of a QFT in Minkowski. Without any assumptions, the vacuum expectation value of the stress tensor is given in terms of an arbitrary tensor function A (x) h0jT (x)j0i =A (x) : (A.1) Let us show how symmetry constraints A (x). Any space-time symmetry of the metric (in this case ) manifests itself in the Hilbert space as an invariance of the vacuum statej0i. For instance, since the Minkowski metric is invariant under space-time translationsx !x +a , the statej0i is invariant under the unitary operator U(a) implementing this symmetry, i.e. U(a)j0i =j0i. Using this in (A.1) it is straightforward to show thatA (x) is independent of the space-time coordinates. In a similar way, invariance of the metric under Lorentz transformation 2 SO(d 1; 1) implies A =h0jU y ()T (x)U()j0i = h0jT ( 1 x)j0i = A : This condition completely xes the tensor A (up to an overall constant) to be equal to the Minkowski metric, i.e. h0jT (x)j0i = a 0 . The vacuum stress tensor is highly constrained in Minkowski due to the fact that it is a maximally symmetry space-time, i.e. it admits a maxi- mal number of d(d + 1)=2 independent Killing vectors. The same is true for QFTs in the other 183 Lorentzian maximally symmetric space-times (anti-)de Sitter, where the vacuum stress tensor is also proportional to the metric. Let us now consider a QFT in AdS 2 S d2 . In this case, the space-time is the product of two maximally symmetric manifolds. Using the Killing vectors in each factor in an analogous way to the Minkowski case, we nd h0jT (x)j0i = 0 B @ a 0 g ij N iA (x) N Ai (x) b 0 g AB 1 C A ; where the indices (i;j) and (A;B) run over AdS 2 andS d2 respectively. Although the o-diagonal contributions N iA (x) are naively not xed by symmetry, they actually vanish. This is because hT iA (x)i 0 has a single index in the AdS 2 direction, meaning it transforms as a vector under its isometries. A non-vanishing vacuum expectation value would have a preferred direction in AdS 2 and be inconsistent with the symmetries of the vacuum. Putting everything together, symmetry considerations alone constraint the vacuum stress tensor of any QFT in AdS 2 S d2 to h0jT (x)j0i =a 0 g ij +b 0 g AB ; (A.2) where a 0 and b 0 are arbitrary constants. For conformal eld theories we can show this explicitly. The vacuum stress tensor for a confor- mally at background is obtained from eq. (21) of [73], that is written in terms of contractions of the Riemann tensor. Instead of using the expression for arbitrary dimensions, let us consider the four dimensional case, which captures the essential features and has the following simple expression hT i 0 /g 1 2 R 2 R 2 + 2R R 4 3 RR : (A.3) To compute this for AdS 2 S d2 we use that the Riemann tensor of any product manifold decom- poses accordingly R =R AdS 2 ijkl +R S d2 ABCD : (A.4) Moreover, since both manifolds are maximally symmetric both terms are proportional to the metric in each factor. Using this in (A.3) we obtain a decomposition of the vacuum stress tensor that is in agreement with (A.2). 184 The result in (A.2) has several interesting consequences. Since the connection also admits a decomposition as the Riemann tensor in (A.4), the covariant derivative of the vacuum stress tensor becomes r hT i 0 =a 0 r g ij +b 0 r g AB =a 0 r k g ij +b 0 r C g AB = 0 : Another important application is obtained by considering a null geodesic x () moving entirely in AdS 2 , i.e. dx =d =dx i =d. Projecting the vacuum stress tensor along this direction gives hT i 0 = dx d dx d hT i 0 =a 0 dx i d dx j d g ij = 0 : Notice that as soon as we consider a null geodesic that also moves in the S d2 direction, this property is no longer true, since a 0 6=b 0 in general. A.2 Light-ray operator algebra In this appendix we show the charges associated to the asymptotic (conformal) Killing vectors at = 0 in (4.28) satisfy the algebra in (4.56) for the CFT in AdS 2 S d2 . We do this using some results of [115] involving the algebra of light-ray operators in the null plane together with a conformal transformation from chapter 2. Let us start by considering thed-dimensional Minkowski metric written in Cartesian coordinates X = (T;X; ~ Y ), so that the null plane XT = 0 is parametrized as in (2.4) X (;~ x ? ) = (;;~ x ? ) ; (;~ x ? )2RR d2 ; (A.5) where (;~ x ? ) are parametrization coordinates, with ane. The light-ray operators on the null plane analogous to those dened at Minkowski null innity are given by b E(~ x ? ) Z +1 1 d b T (;~ x ? ) ; b K(~ x ? ) Z +1 1 d b T (;~ x ? ) ; (A.6) and b N A (~ x ? ) Z +1 1 d b T A (;~ x ? ) = Z +1 1 d dX d dX dx A ? b T (;~ x ? ) ; (A.7) where we have added the hat to dierentiate from the light-ray operators dened previously in (4.32) and (4.33). In [115] it was shown that under some general assumptions these operators 185 satisfy the following algebra 1 h b E(~ x ? 1 ); b E(~ x ? 2 ) i = 0 ; h b K(~ x ? 1 ); b K(~ x ? 2 ) i = 0 ; h b K(~ x ? 1 ); b E(~ x ? 2 ) i =i(~ x ? 12 ) b E(~ x ? 2 ) ; h b N A (~ x ? 1 ); b E(~ x ? 2 ) i =i(~ x ? 12 )@ A b E(~ x ? 2 ) +i@ A (~ x ? 12 ) b E(~ x ? 2 ) ; h b N A (~ x ? 1 ); b K(~ x ? 2 ) i =i(~ x ? 12 )@ A b K(~ x ? 2 ) +i@ A (~ x ? 12 ) b K(~ x ? 2 ) ; h b N A (~ x ? 1 ); b N B (~ x ? 2 ) i =2i(~ x ? 12 )@ [A b N B] +i@ A (~ x ? 12 ) b N B (~ x ? 2 ) +i@ B (~ x ;? 12 ) b N A (~ x ? 1 ) ; (A.8) where (~ x ? 12 ) (d2) (~ x ? 1 ~ x ? 2 ) and derivatives are with the respective arguments. We now use the conformal transformation from section 2.2.1, mapping the Minkowski null plane to AdS 2 S d2 . The mapped surface in AdS 2 S d2 , obtained from equations (2.40) and (2.39), is given by ( + ; ;~ y ) = 0 ; 2 arctan 2p(~ x ? ) +p(~ x ? ) 0 ;~ x ? ; (;~ x ? )2RR d2 ; (A.9) where = andp(~ x ? ) is dened in (2.31). We have also rigidly translated the time coordinate ! + 0 . The surface travels between the two AdS 2 boundaries along the future horizon H + at + = 0 seen in gure 4.1. The conformal factor w 2 (;~ x ? ) relating Minkowski to AdS 2 S d2 in these global coordinates (4.11) was computed in table 2.3 and is given by w 2 (;~ x ? ) =p 2 (~ x ? ) : (A.10) As it is independent of, it implies is also an ane parameter for the surface (A.9) in AdS 2 S d2 . The conformal factor is very much related to the determinant of the induced metric of the surface (A.9) on AdS 2 S d2 , a unit sphere S d2 ds 2 x (;~ x ? ) =d 2 d2 = 4jd~ x ? j 2 (1 +j~ x ? j) 2 =) p h ind = 1 w(~ x ? ) d2 : (A.11) 1 The last commutator is written in a dierent way from the one given in [115], using the following expansion around~ x ? 2 =~ x ? 1 ~ x ? 12 @B(~ x ;? 12 ) b NA(~ x ? 2 ) =@B(~ x ;? 12 ) h b NA(~ x ? 1 ) (x ? 12 ) C @C b NA(~ x ? 1 ) +::: i =@B(~ x ;? 12 ) b NA(~ x ? 1 ) +(~ x ;? 12 )@B b NA(~ x ? 1 ) ; where in the second equality we have integrated by parts and used that quadratic and higher order contributions to the expansion vanish due to the action of the derivative of Dirac's delta. This equivalent way of writing the commutator has the advantage that both sides are explicitly anti-symmetric under the change (A;~ x ? 1 )$ (B;~ x ? 2 ). 186 Using this together with the mapping of the stress tensor under a general conformal transformation (4.34) we nd U b T (;~ x ? )U y = p h ind T (;~ x ? ) ; U b T A (;~ x ? )U y = p h ind T A (;~ x ? ) ; (A.12) where the projection in the (;A) components is with respect to the surface (A.9). This gives a simple way of mapping the light-ray operators in (A.6) and (A.7): we simply replace the hats by bars and multiply by the overall factor p h ind . Applying this to the rst three relations in the light-ray algebra (A.8) is straightforward, where the extra factor of p h ind in the third relation allows us to replace the at space Dirac delta (~ x ? 12 ) by the appropriate one associated to the sphere S d2 , given by (~ x ? 12 ) (~ x ? 12 ) p h ind = 1 p h ind (d2) (~ x ? 1 ~ x ? 2 ) : (A.13) The last three relations in (A.8) require additional care as derivative of the light-ray operators appear on the right-hand side. For instance, applying the adjoint action of U to the rst term on the fourth commutator in (A.8) gives U i(~ x ? 12 )@ A b E(~ x ? 2 ) U y =i(~ x ? 12 )@ A p h ind E(~ x ? 2 ) : (A.14) Further expanding the right-hand side we get an additional term involving the derivative of p h ind ; however, this is compensated by the second term on the fourth commutator in (A.8), where we must replace the at space Dirac delta by the one associated to S d2 in (A.13). The end result is that the structure of the last three relations in (A.8) is preserved under the mapping, so that we obtain the following algebra for the light-ray operators in AdS 2 S d2 h E(~ x ? 1 ); E(~ x ? 2 ) i = 0 ; h K(~ x ? 1 ); K(~ x ? 2 ) i = 0 ; h K(~ x ? 1 ); E(~ x ? 2 ) i =i (~ x ? 12 ) E(~ x ? 2 ) ; h N A (~ x ? 1 ); E(~ x ? 2 ) i =i (~ x ? 12 )D A E(~ x ? 2 ) +i E(~ x ? 2 )D A (~ x ? 12 ) ; h N A (~ x ? 1 ); K(~ x ? 2 ) i =i (~ x ? 12 )D A K(~ x ? 2 ) +i K(~ x ? 2 )D A (~ x ? 12 ) ; h N A (~ x ? 1 ); N B (~ x ? 2 ) i =2i (~ x ? 12 )D [A N B] +i N A (~ x ? 1 )D B (~ x ? 12 ) +i N B (~ x ? 2 )D A (~ x ? 12 ) : (A.15) 187 We have also replaced the ordinary derivatives @ A by covariant derivatives D A on the unit sphere S d2 . While this is trivial for the derivatives acting on scalars, it is also allowed for the ones acting on N A since the anti-symmetric combination means the connection of S d2 does not contribute. We can now construct the analogous charges to (4.31) for the AdS 2 S d2 case T (f) = Z S d2 d (~ x ? )f(~ x ? ) E(~ x ? ) ; R(Y ) = Z S d2 d (~ x ? ) (DY ) d 2 K(~ x ? ) +Y A N A (~ x ? ) ; D(g) = Z S d2 d (~ x ? )g(~ x ? ) K(~ x ? ) ; (A.16) and use (A.15) to compute its algebra, so that we nally nd (4.56). As an example let us write the computation of the second commutator explicitly T (f); R(Y ) =i Z S d2 d (~ x ? ) (DY ) d 2 f(~ x ? ) E(~ x ? ) + Z S d2 d (~ x ? 1 )d (~ x ? 2 )f(~ x ? 1 )Y A (~ x ? 2 ) h E(~ x ? 1 ); N A (~ x ? 2 ) i =i Z S d2 d (~ x ? ) (DY ) d 2 f(~ x ? ) E(~ x ? )i Z S d2 d (~ x ? )Y A (~ x ? )D A f(~ x ? ) =i T ( b f) ; (A.17) where we have been careful with the signs when integratingD A (~ x ? 12 ) and we have dened b f(~ x ? ) as in the second line in (4.56). The rest of the algebra in (4.56) follows from analogous computations. The most involved commutator is the one involving two operators R, where we must be careful with signs and the commutation of covariant derivatives. Finally, we can relate the parametrization coordinates (;~ x ? ) to the space-time coordinates (u;;~ y ) used to described the AdS 2 S d2 metric in (4.11). We can do this using the description of the surface in the global coordinates in (A.9) together with the relations between the coordinates in (4.10), which gives ( + ; ;~ y ) = 0 ; 2 arctan 2p(~ x ? ) +p(~ x ? ) 0 ;~ x ? =) (u;;~ y ) = 1=2 + 2p(~ x ? ) ; 0;~ x ? : (A.18) Using this we can write the light-ray operators in (A.16) in terms of the space-time coordinates (u;;~ y ) as in (4.32)-(4.33). Shifting u! u 1=2 we see that is proportional to u, meaning the only light-ray operator that changes is E(~ x ? )! 2p(~ x ? ) E(~ x ? ), which can be absorbed in the supertranslations charge T (f) in (A.16) by redeningf(~ x ? ). Since the parametrization coordinate ~ x ? is the same as ~ y, the charges (A.16) become (4.31) (without the factor 1= d2 , that is not present for the AdS 2 S d2 case). 188 A.3 Discrete symmetry of quantum theory In this appendix we study discrete symmetries of QFTs in Minkowski and AdS 2 S d2 that arise from the isometries of the Euclidean theory. Let us start by considering a theory dened in the Euclidean plane in Cartesian coordinates (t E ;x 1 ;~ x ) which has a symmetry group given by SO(d). In particular we can consider a rotation in the plane (t E ;x 1 ) given by Euclidean rotation : (t E ;x 1 ;~ x ) ! (t E ;x 1 ;~ x ) : (A.19) Upon analytic continuation t E = it this gives the discrete CRT symmetry given in (4.45). If we change from Cartesian coordinates in Minkowski to spherical, dened as x 1 =r j~ yj 2 1 j~ yj 2 + 1 ; ~ x =r 2~ y j~ yj 2 + 1 ; (A.20) the Minkowski metric becomes ds 2 =dt 2 +dx 2 1 +jd~ xj 2 =dt 2 +dr 2 +r 2 d 2 d2 ; whered d2 is written in stereographic coordinates as in (4.7). The Minkowski CRT transformation (4.45) in these coordinates becomes CRT : (t;r;~ y ) ! t;r; ~ y j~ yj 2 : (A.21) For even space-time dimension we can equivalently consider the CPT symmetry instead, obtained by applying a rotation on the remaining spatial Cartesian coordinates ~ x, so that CRT becomes CPT, i.e. (t;x 1 ;~ x )!(t;x 1 ;~ x ). In terms of the transformation written in spherical coordinates in (A.21), this corresponds to adding an additional minus sign on the ~ y inversion in (A.21). For the unit sphereS d2 the~ y inversion with the minus sign corresponds to the antipodal map, as can be seen by noting the unit vector~ n2R d1 dening the sphere S d2 transforms as CPT : ~ n = j~ yj 2 1 j~ yj 2 + 1 ; 2~ y j~ yj 2 + 1 ! ~ n : It is only for even dimension that we have this CPT transformation, as for odd d the re ection in the Cartesian coordinates ~ x!~ x we started from is not a part of the connected sector of the Euclidean symmetry group. 189 Let us now consider a QFT in AdS 2 S d2 and show the CRT transformation in (A.21) is also a symmetry of the QFT. While for conformal theories this follows after using the space- times are related by a conformal transformation, we want to give a more explicit proof working directly in AdS 2 S d2 . We do this using the embedding space formalism of the conformal group, whose main idea is to embed the space-time of the CFT into a larger space where conformal transformations act linearly. Since the conformal group is isomorphic to SO(d; 2) we dene the embedding coordinates X2R d;2 X = (X 0 ;X i ;X d ;X d+1 ) ; in the space ds 2 =(dX 0 ) 2 + d1 X i=1 (dX i ) 2 + h (dX d ) 2 (dX d+1 ) 2 i : (A.22) Every group element g2 SO(d; 2) has a representation in terms of a matrix g which has a linear action in the embedding coordinates given by ordinary matrix multiplication X 0 = g X. The relation with the d-dimensional space-time is obtained by considering the projective null cone PC = X2R 2;d : (XX) = 0 XcX ; c2R + ; (A.23) where (XX) is computed using the embedding metric (A.22). The denominator means there is a gauge redundancy in the scaling ofX. This gauge freedom can be used to x one of the components in the vector X arbitrarily, which is usually taken as X + =X d +X d+1 = xed : (A.24) This condition is equivalent to xing the conformal frame where thed-dimensional theory is dened. To describe a CFT in the metric AdS 2 S d2 we consider the following parametrization of the projective null cone X = sin() sin() ;~ n; cot(); cos() sin() ; (A.25) where ~ n2 R d1 has unit normj~ nj = 1. It is straightforward to check this vector is null in the embedding space, with gauge xing X + = cos() + cos() sin() : (A.26) 190 A convenient parametrization for the unit vector~ n is obtained by taking stereographic coordinates ~ y2R d2 , so that (A.25) becomes X(;;~ y ) = sin() sin() ; 2y A 1 +j~ yj 2 ; j~ yj 2 1 j~ yj 2 + 1 ; cot(); cos() sin() : (A.27) Computing the induced metric we nd ds 2 ind =dX(;;~ y )dX(;;~ y ) = d 2 +d 2 sin 2 () + 4jd~ yj 2 (1 +j~ yj) 2 ; that is precisely the AdS 2 S d2 metric we are interested in. Let us now consider a rotation in the embedding space between the coordinates (X 0 ;X d1 ) (X 0 ;X d1 ) ! (X 0 ;X d1 ) : (A.28) While this is not a transformation of the Lorentzian conformal group SO(d; 2) it is in the Euclidean group SO(d 1; 1), similarly to the Minkowski case in (A.19). Using this in (A.27) we get ~ X(;;~ y ) = sin() sin() ; 2y A 1 +j~ yj 2 ; j~ yj 2 1 j~ yj 2 + 1 ; cot(); cos() sin() : (A.29) Comparing with (A.27) we see the transformation induced in thed-dimensional coordinates (;;~ y ) is precisely given by the CRT transformation in (4.52). Moreover, since the transformation (A.28) does not change the gauge xing condition in (A.26) the CRT transformation is an exact isometry of AdS 2 S d2 , meaning it is actually a symmetry of the vacuum state of any QFT, not necessarily conformal. A.4 Commutator identity In this appendix we compute the following commutator W;e iV = 1 X n=1 (i) n n! [V n ;W ] : (A.30) To do so, we use induction to prove the following identity [V n ;W ] = n X m=1 (1) m+1 n m V nm L m V (W ) : (A.31) For n = 1 the identity is obviously true, so let us assume it holds for n and show it does for n + 1, where we have V n+1 ;W =V n L V (W ) + n X m=1 (1) m+1 n m V nm L m V (W )V ; 191 and we have already used our hypothesis (A.31). Rearrange the terms to nd V n+1 ;W =V n L V (W ) + n X m=1 (1) m+1 n m V nm VL m V (W )L m+1 V (W ) = (n + 1)V n L V (W ) + n X k=2 (1) k+1 n k V (n+1)k L k V (W ) + n+1 X k=2 (1) k+1 n k 1 V (n+1)k L k V (W ) : where in the rst line we have redened the summation indices k = m and k = m + 1 for each term. Rearranging one more time, we nd V n+1 ;W = (n + 1)V n L V (W ) + n X k=2 (1) k+1 n k + n k 1 V (n+1)k L k V (W ) + (1) n L n+1 V (W ) ; The term between curly brackets is the binomial n+1 k , while the two additional terms give the k = 1 and k =n + 1 contributions so that we recover (A.31) with n + 1. Using this in (A.30) we get the following identity W;e iV = 1 X n=1 n X m=1 n m (i) n (1) m n! V nm L m V (W ) ; (A.32) that we apply in our analysis in (4.62). A.5 Bulk superrotation integral curves The integral curves associated to bulk vector R (Y ) at = 0 (4.104) are determined by the following system of equations u 0 (s) = Y 0 ((s)) 1 +x(s) 2 u(s) ; 0 (s) =Y ((s)) ; x 0 (s) =Y 0 ((s))x(s) ; (A.33) where s is the parameter along the curve. We must solve for the three functions (u(s);(s);x(s)) with initial conditions such that s = 1 gives the identity. The second dierential equation can be written in terms of a simple integral of Y () s = Z (s) d 0 Y ( 0 ) =) (s)(;s) : (A.34) 192 For any well behaved function Y () the integral can be solved and inverted so that the function (;s) is written explicitly. 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Abstract (if available)

Abstract The first part of this thesis explores the interesting failure of the classical energy conditions when incorporating quantum effects through the semi-classical approximation. Our main focus is on the achronal Averaged Null Energy Condition (ANEC), which provides a natural extension of the classical Null Energy Condition. We give several independent proofs of the achronal ANEC for different classes of QFTs defined on certain curved backgrounds, providing concrete evidence that points towards the achronal ANEC as a fundamental feature of semi-classical gravity. As an application, we study its consequences on the infinite dimensional BMS algebra that arises on certain black holes horizons and asymptotically flat space-times. ? In the second part of the thesis we study two dimensional theories of quantum gravity, where several of the obstacles that arise when trying to quantize gravity in higher dimensions are absent. Recent progress has shown a class of Jackiw-Teitelboim gravity theories can be quantized explicitly and observables computed in a perturbative topological expansion to arbitrary order. Remarkably, the topological expansion is exactly reproduced by certain ensembles of random matrices. We show how the matrix model description can be used to provide a fully non-perturbative definition of the quantum gravity theory that allows for concrete and explicit calculations of observables beyond perturbation theory.

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Asset Metadata

Creator Rosso, Felipe (author)

Core Title Explorations in semi-classical and quantum gravity

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Degree Conferral Date 2021-08

Publication Date 07/16/2021

Defense Date 05/17/2021

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag ANEC,averaged null energy condition,BMS,Bondi,energy conditions,entanglement,gravity,high energy theory,Jackiw-Teitelboim gravity,Jackiw-Teitelboim supergravity,JT gravity,JT supergravity,low dimensional gravity,matrix models,Metzner and Sachs,modular Hamiltonian,OAI-PMH Harvest,Physics,quantum,random matrices,semi-classical,van der Burg

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Johnson, Clifford V. (committee chair), Bars, Itzhak (committee member), Malikov, Fedor (committee member), Pilch, Krzysztof (committee member), Zanardi, Paolo (committee member)

Creator Email felipero@usc.edu,feliperosso6@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC15580170

Unique identifier UC15580170

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Document Type Dissertation

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Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

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