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Generalizations of holographic renormalization group flows

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Content GENERALIZATIONS OF HOLOGRAPHIC RENORMALIZATION GROUP FLOWS by Minwoo Suh A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS) August 2012 Copyright 2012 Minwoo Suh Acknowledgements Along the way of research under my advisor, Krzysztof Pilch’s guidance, there were moments of excitement and delight. I am grateful to Krzysztof for his teachings while the research presented in this dissertation was performed. He introduced me to the field of supergravity and string theory. He demonstrated himself to me the mathematical rigor, logical clarity and other principles of science research. In the time of research hardship, it was his encouragement, humour with good portion of sarcasm, and abundant patience that made me to drive on. It is a pleasure to thank Nicholas P. Warner for sharing his knowledge and experience for the research presented in this dissertation. I have greatly benefited from his courses on general relativity and other topics. I am grateful to Krzysztof Pilch, Nicholas P. Warner, Itzhak Bars, Clifford V . John- son and Dennis Nemeschansky from whom I have learned much in the group meetings and in the courses they gave. I would like to thank Krzysztof Pilch, Nicholas P. Warner, Itzhak Bars, Robin Shake- shaft and Fedor Malikov for serving in my qualifying and dissertation committee and for providing me with encouragement that I needed. I would like to thank my fellow students. I thank Nikolay Bobev for sharing his knowledge on physics and guiding me through by his experience. I really enjoyed the dinners with Nikolay Bobev, Elena Bobeva, Bruna de Oliveira and Toby Jacobson in the early years of study and they are unforgettable part of life here. I thank Isaiah Yoo for enthusiastic conversations on physics and our lives in the days of study. It was pleasure to have Veselin Filev, Arnab Kundu, Tameem Albash, Nikolay Bobev, Dima Rychkov, Isaiah Yoo, Orestis Vasilakis, Ben Niehoff and Scott Macdonald as office ii mates. Conversations on physics and chatters with them were always encouraging and fun and made the office to be a more inhabitable place. It is a pleasure to thank the members of Power of Praise Church, Pastor Seungho Synn and his wife, Nackkum Paik. I enjoyed friendships in Him first time in life and they lead me to walk closer to Him. Literally I cannot thank enough my father, Nam Joon Suh, and my mother, Hai In Lee. With belief, hope and love they brought me to the place where I could start this journey. I thank my sister, Inji’s family. The huge ocean between us brought us even closer. I thank Him. iii Table of Contents Acknowledgements ii Abstract vi Chapter 1: Introduction 1 1.1 The holographic principle . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Holographic renormalization group flows . . . . . . . . . . . . . . 2 1.3 Example: TheN = 1 supersymmetric RG flow . . . . . . . . . . . . 7 1.4 Generalizations of holographic RG flows . . . . . . . . . . . . . . . 11 Chapter 2: Holographic RG flows with dilaton and axion fields 14 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 The Janus solutions in supergravity . . . . . . . . . . . . . . . . . . 18 2.2.1 The original Janus solution . . . . . . . . . . . . . . . . . . 18 2.2.2 The super Janus . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.3 The Janus solutions in type IIB supergravity . . . . . . . . . 21 2.3 Truncation ofN = 8 gauged supergravity in five dimensions . . . . 24 2.3.1 TheSU(3)-invariant truncation . . . . . . . . . . . . . . . . 24 2.3.2 The supersymmetry equations . . . . . . . . . . . . . . . . 28 2.3.3 The numerical solutions . . . . . . . . . . . . . . . . . . . 37 2.4 Super Janus inN = 2 gauged supergravity in five dimensions [23] 39 2.5 Lift of theSU(3)-invariant truncation to type IIB supergravity . . . 40 2.5.1 The metric . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.5.2 The dilaton/axion fields . . . . . . . . . . . . . . . . . . . . 42 2.6 Supersymmetric Janus solution in type IIB supergravity [20] . . . . 44 2.7 Lift of theSU(3)-invariant truncation to type IIB supergravity (con- tinued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.7.1 The three-form flux . . . . . . . . . . . . . . . . . . . . . . 46 2.7.2 The five-form flux . . . . . . . . . . . . . . . . . . . . . . . 48 2.8 Type IIB supergravity on Sasaki-Einstein manifolds [50, 51] . . . . 51 2.9 TheSU(2)U(1)-invariant flows with dilaton and axion fields . . . 53 2.9.1 The supersymmetry variations (I) . . . . . . . . . . . . . . . 56 2.9.2 The supersymmetry variations (II) . . . . . . . . . . . . . . 57 Chapter 3: Holographic RG flows with gauge fields 60 3.1 TheSU(3)-invariant flow with electric potentials . . . . . . . . . . 60 3.1.1 The supersymmetry variations . . . . . . . . . . . . . . . . 60 3.1.2 The flow equations without dilaton and axion fields . . . . . 64 iv 3.1.3 The flow equations with dilaton and axion fields . . . . . . . 67 3.2 TheSU(3)-invariant flow with magnetic fields . . . . . . . . . . . . 68 3.2.1 The magnetic brane solutions . . . . . . . . . . . . . . . . . 68 3.2.2 Configurations of magnetic fields . . . . . . . . . . . . . . . 71 3.2.3 TheSU(3)-invariant flow with magnetic fields (case I) . . . 72 3.2.4 TheSU(3)-invariant flow with magnetic fields (case II) . . . 76 Chapter 4: Conclusions 78 Appendix A:N = 8 gauged supergravity in five dimensions 79 Appendix B:SU(2; 1) algebra 82 Appendix C: The supersymmetry variations for spin-1/2 fields 84 Appendix D: The supersymmetry variations for spin-3/2 fields 86 Appendix E: The parametrizations of the scalar manifold 88 Appendix F: The field equations of theSU(3)-invariant truncation 91 Bibliography 93 v Abstract The AdS/CFT correspondence conjectures the duality between type IIB supergravity on AdS 5 S 5 andN = 4 super Yang-Mills theory. Mass deformations ofN = 4 super Yang-Mills theory drive renormalization group (RG) flows. Holographic RG flows are described by domain wall solutions interpolating between AdS 5 geometries at critical points ofN = 8 gauged supergravity in five dimensions. In this thesis we study two directions of generalizations of holographic RG flows. First, motivated by the Janus solutions, we study holographic RG flows with dilaton and axion fields. To be specific, we consider theSU(3)-invariant flow with dilaton and axion fields, and discover the known supersymmetric Janus solution in five dimensions. Then, by employing the lift ansatz, we uplift the supersymmetric Janus solution of the SU(3)-invariant truncation with dilaton and axion fields to a solution of type IIB super- gravity. We identify the uplifted solution to be one of the known supersymmetric Janus solution in type IIB supergravity. Furthermore, we consider theSU(2)U(1)-invariant N = 2 andN = 1 supersymmetric flows with dilaton and axion fields. Second, motivated by the development in AdS/CMT, we study holographic RG flows with gauge fields. We consider the SU(3)-invariant flow with electric potentials or magnetic fields, and find first-order systems of flow equations for each case. vi Chapter 1 Introduction 1.1 The holographic principle With the great success of 20th century physics, quantum field theory and general rel- ativity, it is one of the final goals of physics: the unification of quantum field theory and general relativity toward theory of quantum gravity. Around the huge success of quantum field theory in the standard model of particle physics in 1970s, some crucial elements for quantum gravity were discovered: supersymmetry, extra dimensions and string theory. Based upon them, in 1980s, it turned out that there are five kinds of string theories: type I, type IIA, type IIB, heterotic O and heterotic E string theories. How- ever, in 1995, it was shown that the five string theories are merely low energy effective theories of more fundamental, but so far unknown, theory named M-theory. Another important element toward quantum gravity was found in the study of black holes by Hawking and Bekenstein in 1980s: The entropy of a black hole is proportional to its surface area. This suggests that the physics of (d+1)-dimensional bulk of a black hole is governed by the physics of the d-dimensional boundary of the black hole. In early 1990s, this idea was expanded to the holographic principle: Quantum field the- ory in d-dimensions is related to quantum gravity in (d+1)-dimensions. This principle was qualitative initially, however, in 1997, Juan Maldacena suggested the first concrete example of the holographic principle: the AdS/CFT correspondence. The AdS/CFT correspondence [1, 2, 3] conjectures the duality between specific kinds of quantum field theory and quantum gravity. The quantum field theory in this 1 case isN = 4 super Yang-Mills theory: This is the unique quantum field theory with maximal supersymmetry and conformal symmetry in four dimensions, hence it is a con- formal quantum field theory (CFT). The gravity theory in this case is type IIB string theory. The low energy effective theory of type IIB string theory is type IIB supergravity which is the unique chiral maximally supersymmetric supergravity in ten dimensions. Type IIB supergravity has vacua involving five-dimensional anti-de Sitter (AdS) space- time. The AdS/CFT correspondence means, even thoughN = 4 super Yang-Mills the- ory and type IIB supergravity are very different theories, there is duality between them, therefore, when we calculate physical quantities in one theory, we would get identi- cal answers from the calculations in the other theory. To be specific, via the AdS/CFT correspondence, the strongly coupled regime of one theory corresponds to the weakly coupled regime of the other theory. Hence, the AdS/CFT correspondence is a useful tool to consider the strongly coupled regime of a theory which is usually hard to study. The AdS/CFT correspondence originally involves theories with supersymmetry and confor- mal symmetry, however, later, it was generalized to theories without them, hence, called the gauge/gravity dualtiy. 1.2 Holographic renormalization group flows The AdS/CFT correspondence [1, 2, 3] conjectures a duality between type IIB string theory on AdS 5 S 5 andN = 4 super Yang-Mills theory (SYM). In this section, we consider the RG flows fromN = 4 SYM, and discuss how they can be studied by solutions of type IIB supergravity via the AdS/CFT correspondence. First, we consider the RG flow from the field theory side [4, 5]. We can deformN = 4 SYM by introducing mass terms to some of the chiral adjoint superfields. The mass deformation breaks the conformal invariance ofN = 4 SYM and drives an RG flow. The 2 RG flow leads to a deformed theory where the conformal invariance is recovered.N = 4 SYM theory and the deformed theory correspond to the ultraviolet (UV) and infrared (IR) fixed points, respectively, and are both conformal field theories. However, along the RG flow, the conformal invariance is broken. Via the AdS/CFT correspondence, RG flows in field theory correspond to certain solutions in gravity theory, the gravity duals, which have identical physics of RG flows. Regarding the RG flows fromN = 4 SYM, the UV and IR fixed points correspond to AdS 5 solutions of type IIB supergravity. There are many ways to check this cor- respondence, and one of the simplest is to compare their symmetries: CFT d has the same symmetry as AdS d+1 , SO(2;d). Hence, the gravity duals of RG flows i:e: the holographic RG flows, [6, 7, 8] can be described by domain wall solutions interpolating between twoAdS 5 geometries. It has not been proved, but with abundant evidence, it is believed that type IIB supergravity compactified onS 5 givesN = 8 gauged supergravity in five dimensions. 1 Hence, studyingN = 8 gauged supergravity in five dimensions should give the equiva- lent physics from studying type IIB supergravity. TheSO(6) gaugedN = 8 supergrav- ity [12, 13, 14] is a maximally supersymmetric gauged supergravity in five dimensions. This theory has a scalar potential from 42 scalar fields living on the scalar manifold, E 6(6) =USp(8). There are vacua,i:e:AdS 5 solutions, at each critical point of the scalar potential. The known critical points ofN = 8 gauged supergravity that are invariant at least underSU(2) U(1) are listed in table 1.1 [16]. Via the AdS/CFT correspondence, holographic RG flows are described by domain wall solutions interpolating between the critical points ofN = 8 gauged supergravity in 1 First, we can compare symmetries and spectrum of type IIB supergravity onS 5 with those ofN = 8 gauged supergravity in five dimensions. Second, there are solutions of type IIB supergravity uplifted from solutions ofN = 8 gauged supergravity in five dimensions. For instance, the holographic RG flows in [9, 10, 11] are the examples. However, it should be noted that not all solutions of type IIB supergravity have their origin inN = 8 gauged supergravity in five dimensions. 3 five dimensions [6, 7, 8]. In this picture, the UV fixed point is the maximally supersym- metric SO(6)-invariant vacuum ofN = 8 gauged supergravityin five dimensions, and IR fixed points are vacua with less supersymmetry and global symmetry. Along the RG flows, some supersymmetry and global symmetry are preserved. Points Gauge symmetry Cosmological constant Supersymmetry c IR =c UV (i) SO(6) 3 4 g 2 N = 8 1 (ii) SO(5) 3 5=3 8 g 2 N = 0 2 p 2 3 (iii) SU(3) 27 32 g 2 N = 0 16 p 2 27 (iv) SU(2)U(1)U(1) 3 8 25 2 1=3 g 2 N = 0 4 5 (v) SU(2)U(1) 2 4=3 3 g 2 N = 2 27 32 Table 1.1 Known critical points ofN = 8 gauged supergravity in five dimensions [16]. In theSO(6) representation, the 42 scalar fields branch as 1 + 1 + 10 + 10 + 20 0 ; (1.1) where the two singlets are the dilaton and axion fields. Via the AdS/CFT correspon- dence, the representations in (1.1) correspond to the gauge coupling, the -angle, the fermion bilinear operators, and the scalar bilinear operator ofN = 4 SYM, respectively [6, 7]. The latter have the form Tr ( i j ); Tr ( i j ); Tr (X a X b ) 1 6 ab Tr (X c X c ); (1.2) wherei,j = 1,::: , 6 anda,b = 1,::: , 4. Holographic RG flows are, hence, obtained by turning on the scalar fields which are dual to the mass deformation operators inN = 4 SYM. 4 The RG flows to the nonsupersymmetricSU(3) andSO(5) critical points were the first examples by Girardello, Petrini, Porrati and Zaffaroni [6] and by Distler and Zamora [7]. We list some of the known RG flows by the number of massive fermion bilinears. One massive fermion bilinear:N = 1 flow It corresponds to theN = 1 supersymmetric RG flow to theN = 2 supersymmetric SU(2)U(1) critical point. It involves two scalar fields, and, dual to fermion bilinear and scalar bilinear, respectively, Tr ( 4 4 ) ! ; 4 X j=1 Tr (X j X j ) 2 6 X j=5 Tr (X j X j ) ! : (1.3) TheN = 1 flow corresponds to the phase discovered by Leigh and Strassler [4], and, is known as the LS flow. TheSU(2) U(1) critical point in supergravity was discovered by Khavaev, Pilch and Warner [16]. The holographic RG flow was studied by Freedman, Gubser, Pilch and Warner, hence, is also known as the FGPW flow from the supergravity aspect [8]. Later, the flow was uplifted to type IIB supergravity by Pilch and Warner [10]. We are not always lead to an IR critical point by RG flows. There are flows which lead the scalar fields to divergences, flows to Hades. However, there are examples that these five-dimensional singularities are overcome when the flow solutions are uplifted to type IIB supergravity [9, 10]. Below are the examples. 5 Two massive fermion bilinears:N = 2 flow It involves two scalar fields, and , dual to a fermion bilinear and a scalar bilinear, respectively, Tr ( 3 3 + 4 4 ) ! ; 4 X j=1 Tr (X j X j ) 2 6 X j=5 Tr (X j X j ) ! : (1.4) TheN = 2 flow was studied and then uplifted to type IIB supergravity [9]. It describes the Coulomb branch ofN = 4 SYM. Three massive fermion bilinears:N = 1 flow Minimally, it involves two scalar fields,m and, dual to a fermion bilinear and a gaugino condensate, respectively, 3 X a=1 Tr ( a a ) ! m; Tr ( 4 4 ) ! : (1.5) The vacua ofN = 1 theories were extensively studied from the field theory aspecte:g: references in [10]. The holographicN = 1 flow was first studied by Girardello, Petrini, Porrati and Zaffaroni, and known as GPPZ flow [17]. Later, it was revisited by Pilch and Warner with more general scalar fields, and then uplifted to type IIB supergravity [10], however, the full uplift of this flow is not known. 6 1.3 Example: TheN = 1 supersymmetric RG flow One of the main technical issues in the study of holographic RG flows is to manage the complexity from the 42 noncompact scalar fields ofN = 8 gauged supergravity in five dimensions. Each vacuum corresponds to a critical point of the scalar potential, however, handling the scalar potential with all 42 scalar fields is not practical. So it turned out to be convenient to truncateN = 8 gauged supergravity to its subsector with global symmetry smaller than SO(6) [16]. In this manner, the RG flows with SU(3) andSU(2) U(1) invariance have been studied [6, 7, 8, 17, 9, 10, 18]. As a specific example, let us consider theN = 1 supersymmetric SU(2)U(1)- invariant flow [8] in (1.2). We set the gauge fields to vanish. The bosonic part of the Lagrangian of theSU(2)U(1)-invariant truncation is e 1 L = 1 4 R + 3@ @ + 1 2 @ @ +P 1 4 e 4 F F +L CS : (1.6) The superpotential is W = 1 4 2 h cosh(2) ( 6 2) (3 6 + 2) i ; (1.7) and the scalar potential is obtained by P = g 2 8 @W @' j 2 g 2 3 jWj 2 ; (1.8) where = e , and ' j are properly normalized fields, ' 1 = , ' 2 = p 6. The scalar potential has three critical points: the maximally supersymmetricSO(6)-invariant point, theN = 2 supersymmetric SU(2)U(1)-invariant point, and the nonsupersymmetric SU(3)-invariant point. In figure 1.1, points 2 and 3 areZ 2 equivalent SU(3)-invariant points and points 4 and 5 areZ 2 equivalentN = 2 supersymmetric points. 7 -0.2 -0.1 0 0.1 0.2 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 2 3 4 5 -0.2 -0.1 0 0.1 0.2 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 Figure 1.1: The contour map of the scalar potential, P , (left) and the superpotential, W , (right), with on the vertical axis and on the horizontal axis [8]. Now we consider the domain wall solution which preserves the Poincar´ e invariance in four dimensions [8], ds 2 = e 2U(r) dx dx dr 2 ; (1.9) where is a Minkowski metric andr is a radial direction corresponding to the energy scale in dual field theory. By having the supersymmetry variations of fermionic fields, i:e: the spin-3/2 and spin-1/2 fields, vanish, we obtain the RG flow equations, d' j dr = g 2 @W @' j ; (1.10) dU dr = g 3 W; (1.11) whose solution interpolates between the critical point with maximal supersymmetry and theN = 2 supersymmetric critical point. A numerical solution of the steepest descent equations is shown on the contour plot of W in figure 1.1. Along the flowN = 1 supersymmetry is preserved. 8 N = 8 gauged supergravity in five dimensions is believed to be a consistent trunca- tion of type IIB supergravity onS 5 ,i:e: solution ofN = 8 gauged supergravity in five dimensions can be uplifted to a solution of type IIB supergravity. There are proposed consistent truncation ans¨ atse for type IIB supergravity fields,i:e: metric, dilaton/axion fields, three- and five-form fluxes. A consistent truncation ansatz for metric was pro- posed in [16], for dilaton/axion fields in [9], and for three- and five-form fluxes in [18]. Employing those ans¨ atse, theN = 1 andN = 2 supersymmetricSU(2)U(1)-invariant flows were uplifted to type IIB supergravity in [10] and [9], respectively. Now we briefly present the uplift of theN = 1 supersymmetric SU(2)U(1)- invariant flow [10]. The IIB dilaton and axion fields are trivial for this flow. The IIB metric is ds 2 = 2 ds 2 1;4 + ds 2 5 ; (1.12) where ds 2 1;4 is an arbitrary solution ofN = 8 gauged supergravity in five dimensions. The internal space metric is ds 2 5 = a 2 2 sech (dx I Q 1 IJ dx J ) + a 2 2 sinh tanh 3 (x I J IJ dx J ) 2 ; (1.13) whereQ IJ is a diagonal matrix withQ 11 =::: =Q 44 =e 2 ,Q 55 =Q 66 =e 4 ,J IJ is an antisymmetric matrix withJ 14 =J 23 =J 65 = 1, and 2 =x I Q IJ x J . The warp factor is 2 = cosh: (1.14) We define complex coordinates corresponding toJ IJ , u 1 = x 1 + ix 4 ; u 2 = x 2 + ix 3 ; u 3 = x 5 ix 6 ; (1.15) 9 and then 0 @ u 1 u 2 1 A = cos g ( 1 ; 2 ; 3 ) 0 @ 1 0 1 A ; u 3 = e sin; (1.16) where g ( 1 , 2 , 3 ) is anSU(2) invariant matrix in terms of Euler angles. The three-form flux is given by [10] F (3) = dA (2) ; (1.17) where the two-form gauge potential is A (2) = C (2) iB (2) ; (1.18) andC (2) andB (2) are RR and NSNS two-form gauge potentials, respectively. We have A (2) = e i (a 1 da 2 3 a 3 d)^ ( 1 i 2 ); (1.19) where a 1 = 2 g 2 tanh cos; (1.20) a 2 = 1 g 2 6 tanh X 1 cos 2 sin; (1.21) a 3 = 2 g 2 tanh X 1 cos 2 sin; (1.22) with X 1 = cos 2 + 6 sin 2 ; (1.23) and i ;i = 1, 2, 3, are theSU(2)-invariant one-forms. 10 The five-form flux is given by [10] F (5) =F +F; (1.24) where F = ! r dr^ dx 0 ^ dx 1 ^ dx 2 ^ dx 3 + ! d^ dx 0 ^ dx 1 ^ dx 2 ^ dx 3 ; (1.25) and ! r = g 8 e 4U cosh 2 4 (cosh(2) 3) cos 2 + 6 (2 6 sinh 2 sin 2 + cos(2) 3) ; (1.26) ! = 1 8 e 4U 1 2 2 cosh 2 + 6 (cosh(2) 3 sin(2): (1.27) 1.4 Generalizations of holographic RG flows In this introduction we briefly considered the RG flows fromN = 4 SYM and their holographic description fromN = 8 gauged supergravity in five dimensions. In this section, we consider some generalizations of holographic RG flows. To understand the first generalization, we consider a class of solutions in type IIB supergravity called the Janus solutions. Via the AdS/CFT correspondence, the only two scalar fields in type IIB supergravity, the dilaton and axion fields, andC (0) , are dual to the gauge coupling and-angle inN = 4 SYM, respectively. Unlike other solutions of type IIB supergravity, the Janus solutions have nontrivial profile of the dilaton field. To be specific, the Janus solutions are characterized by two main features: (i) they are AdS-domain wall solutions with an interface, (ii) the dilaton field takes constant values on both sides of the interface, but it jumps across the interface. As the dilaton field 11 varies, the gauge coupling of the dual gauge theory varies across the interface, i.e. the dual gauge theories are defect conformal field theories. The dual gauge theory isN = 4 SYM in 3+1 dimensions with a 2+1 dimensional interface. The holographic RG flows discussed so far have only involved the scalar fields dual to the fermion or scalar bilinear operators, but not the singlets in (1.1) which are dual to the five-dimensional dilaton and axion fields. Motivated by the Janus solutions we study the holographic RG flows with dilaton and axion fields. Specifically we will concentrate on theSU(3)-invariant flow [11], and will discover that this flow solution involving the dilaton and axion fields indeed reproduces the known Janus solutions with SU(3)-invariance inN = 2 gauged supergravity [23] and in type IIB supergravity [20]. Furthermore, we will consider theSU(2)U(1)-invariantN = 1 andN = 2 supersym- metric RG flows with dilaton and axion fields, however, unlike theSU(3)-invariant flow, it appears that they cannot involve nontrivial dilaton and axion fields. To consider the second kind of generalization, we briefly discuss the recent develop- ment in applying the AdS/CFT correspondence to condensed matter physics: AdS/CMT. One of the obstacles in condensed matter physics is that the interesting condensed mat- ter systems are usually described by strongly coupled field theories. On the other hand, the AdS/CFT correspondence provides an effective tool to study strongly coupled field theories through weakly coupled gravity theories. This AdS/CMT was initiated by phe- nomenological models in gravity theories which exhibit some properties of interesting condensed matter systems, e:g: superconductors, Fermi liquids, and magnetism. One of the popular phenomenological models is the Abelian Higgs model which involves a metric, scalar fields with nontrivial scalar potential, and gauge fields [24, 25, 26]. Holo- graphic superconductors were constructed as electrically charged black hole solutions of this model that develop scalar hair below a critical temperature. On the other hand, there are also top-down models of AdS/CMT from supergravity theories. Unlike the 12 phenomenological models, they provide the precise dual field theories. For instance, holographic superconductors were constructed in type IIB supergravity [27, 28] and in d = 11 supergravity [29, 30]. Also magnetically charged brane solutions were studied in various supergravity theories [31, 32, 33, 34, 35, 36]. Due to the top-down models of AdS/CMT, consistent truncation involving gauge fields has become an interesting topic. The holographic RG flows discussed so far have involved only the scalar fields, and not the gauge fields. Recently, however, motivated by AdS/CMT models with electric potentials, theSU(2)U(1)-invariantN = 1 supersymmetric RG flow [8, 10] in (1.2) was generalized to involve electric potentials, and a flow interpolating between two globalAdS 5 was discovered [37]. In the same spirit, we will study electrically charged SU(3)-invariant flow. Furthermore, we will also study magnetically charged SU(3)- invariant flows. The plan for this thesis is as follows. In chapter 2 we study the generalization of holographic RG flows to involve the dilaton and axion fields. From section 2.1 to sec- tion 2.8 we concentrate on theSU(3)-invariant truncation with dilaton and axion fields based on the paper [11]. In section 2.9 we, further, consider theSU(2)U(1)-invariant flows with dilaton and axion fields. In chapter 3 we study holographic RG flows involv- ing electric potentials or magnetic fields in section 3.1 and section 3.2, respectively. Conclusions are presented in chapter 4. Technical details are collected in appendices. 13 Chapter 2 Holographic RG flows with dilaton and axion fields 2.1 Introduction The Janus solutions provide a class of examples for the AdS/CFT correspondence [1]. The Janus solutions are characterized by two main features: (i) they are AdS-domain wall solutions with an interface, (ii) the dilaton field takes constant values on both sides of the interface, but it jumps across the interface. As the dilaton field is not constant, the coupling constant of the dual gauge theory varies across the interface, i.e. the dual gauge theories are defect conformal field theories. The first example of Janus solutions was discovered in type IIB supergravity with no supersymmetries by Bak, Gutperle and Hirano in [19]. The dual gauge theory isN = 4 super Yang-Mills theory in 3+1 dimensions with a 2+1 dimensional interface. Even though this solution breaks all the supersymmetries, the stability against a large class of perturbations was proved in [19, 38]. After the discovery of the original Janus solution, the dual gauge theory was stud- ied in [39]. It was observed that by reducing SO(6) R-symmetry of the dual gauge theory down to at least SU(3), some supersymmetries were restored. Motivated by this observation, Clark and Karch constructed a supersymmetric Janus solution with 14 SU(3) isometry, super Janus [23], based on the studies of curved domain wall solu- tions [40, 41, 42, 43, 44] inN = 2 gauged supergravity with one hypermultiplet in five dimensions [45, 46]. Later, Janus gauge theories were constructed more systematically in [47]. It gives the complete classification of all possible Janus solutions in type IIB supergravity. According to the classification, there are four kinds of solutions with SO(6), SU(3), SU(2)U(1) and SO(3)SO(3) isometries, and each of them has zero, four, eight, and sixteen Poincar´ e supersymmetries, respectively. Among these, the Janus solution with no supersymmetry is the original Janus solution [19]. By D’Hoker, Estes and Gut- perle, the Janus solutions with four and sixteen supersymmetries were constructed in type IIB supergravity in [20] and [21, 22], respectively. Later, the Janus field theories in [47] were generalized to allow the theta-angle to vary which is holographicallly dual to the axion field, and were also applied to construct three-dimensional Chern-Simons theories withN = 4 supersymmetries in [48]. Despite of all these developments in Janus geometries, as the five- and ten- dimensional solutions were constructed independently, the relation between those solu- tions are far from obvious. However, asN = 2 gauged supergravity with one hyper- multiplet is a truncation ofN = 8 gauged supergravity in five dimensions [12, 13, 14], it was conjectured by Clark and Karch in [39] that the super Janus inN = 2 gauged supergravity could be embedded inN = 8 gauged supergravity in five dimensions. If this embedding could be achieved, as partial results of lift for embeddingN = 8 gauged supergravity to type IIB supergravity onS 5 are readily known [16, 9, 10], one should be able to uplift the supersymmetric Janus solution in five dimensions to the one in type IIB supergravity. This will provide us with the bridge between the known supersymmetric Janus solutions in five and ten dimensions. In this section, we indeed show that the super 15 Janus can be embedded inN = 8 gauged supergravity in five dimensions and its uplift gives the supersymmetric Janus solution in type IIB supergravity [20]. In order to address these questions, we will revisit theSU(3)-invariant truncation of N = 8 gauged supergravity in five dimensions which was studied in [14] and [6, 7, 17]. Later it was uplifted to type IIB supergravity in [10]. However, in these studies, there was only one real scalar field in the flat domain wall, and the dilaton/axion fields were suppressed. In order to construct Janus solutions, we will generalize the previous studies in two aspects: (i) we extend the field content to include the dilaton/axion fields, so we will have two complex or four real scalar fields, (ii) we consider theAdS-domain wall instead of the flat domain wall. However, as it was known inN = 2 gauged supergravity in five dimensions in [40, 43], we will find that the two directions of generalization are in fact equivalent, i.e. one can turn on the dilaton/axion fields only in the curved background, and vice versa. Finally we will show that theSU(3)-invariant truncation with the dilaton/axion fields indeed has a solution identical to the super Janus in [23]. Then we will uplift the solution of theSU(3)-invariant truncation to type IIB super- gravity by employing the consistent truncation ansatz for metric and dilaton/axion fields in [16, 9, 10]. Though there are the lift formulae for three- and five-form fluxes proposed in [18], we find that they do not work for the curved domain walls. We propose modified lift formulae similar to those of [18] for three- and five-form fluxes, and check that they generate correct fluxes for the cases we are considering. Finally we will show that the lift of theSU(3)-invariant truncation indeed falls into a special case of the supersymmetric Janus solution in type IIB supergravity in [20]. Of independent interest from the Janus solutions, there has been notable develop- ment in consistent truncation of type IIB supergravity on Sasaki-Einstein manifolds recently [50, 51, 52, 53]. We will show that the lift of theSU(3)-invariant truncation to type IIB supergravity provides a particular example of the truncation in [50, 51]. 16 Furthermore, we study the SU(2)U(1)-invariant flows with dilaton and axion fields. We will find that, unlike the SU(3)-invariant truncation, the dilaton and axion fields are trivial in theN = 2 supersymmtric flow. In section 2.2 we review the Janus solutions in supergravity. In section 2.3 we begin by studying theSU(3)-invariant truncation ofN = 8 gauged supergravity in five dimensions with dilaton and axion fields. In section 2.4 we show that a solution of the SU(3)-invariant truncation is identical to the super Janus inN = 2 supergravity in five dimensions. In section 2.5 we lift the solution of theSU(3)-invariant truncation to type IIB supergravity by employing consistent truncation ansatz for metric and dilaton/axion fields. In section 2.6 we show that the lifted metric and dilaton/axion fields completely fix the supersymmetric Janus solution with SU(3) isometry in type IIB supergravity. In section 2.7 we continue the lift of theSU(3)-invariant truncation for three- and five- form fluxes. In section 2.8 we consider the consistent truncation of type IIB supergravity on Sasaki-Einstein manifolds in relation with theSU(3)-invariant truncation. In section 2.9 we study theSU(2)U(1)-invariant flows with dilaton and axion fields. In appendix A we briefly reviewN = 8 gauged supergravity in five dimensions. In appendix B the SU(2; 1) algebra is presented. In appendix C details of the supersymmetry variation for spin-3/2 fields are presented for theSU(3)-invariant truncation. Appendix D summa- rizes the different parametrizations of the scalar manifold in this paper. In appendix E we present the field equations in five dimensions. 17 2.2 The Janus solutions in supergravity In this section we review the Janus solutions in supergravity. 2.2.1 The original Janus solution The original Janus solution [19] in type IIB supergravity is an asymptotically AdS 5 space with a spatially varying dilaton. The original Janus solution has a metric, a dilaton, and a five-form flux with the other fields vanishing. The metric takes the form ofAdS 4 - slicedAdS 5 , ds 2 =f(r)ds 2 AdS 4 dr 2 +ds 2 S 5: (2.1) The dilaton field and the five-form flux are, respectively, = (r); (2.2) F (5) = 2f(r) 1=2 dr^! AdS 4 + 2! S 5; (2.3) where! is the unit volume form for the respective space. When one solves the equations of motion, one finds that the dilaton field takes constant values at the boundaries, but it jumps across an interface on the coordinater. Due to this nontrivial profile of the dilaton field, this solution is named as Janus solution. This solution breaks all supersymmetries, but the stability against a large class of perturbations has been proved in [19, 38]. The dual gauge theory is a 3+1 dimensional gauge theory with a 2+1 dimensional planar interface. The gauge theory on each side of the planar interface isN = 4 super Yang-Mills theory, and the gauge coupling varies discontinuously across the interface. Via the AdS/CFT correspondence,e = g 2 YM 4 , where is the dilaton field of type IIB supergravity andg YM is the coupling constant ofN = 4 super Yang-Mills theory. Hence, as the dilaton field varies, the gauge coupling in dual field theory varies. 18 2.2.2 The super Janus We briefly reviewN = 2 gauged supergravity with one hypermultiplet in five dimensions [45, 46]. The bosonic sector of the theory has a graviton e a , a vector field A , and four scalar fieldsq X . The scalar fields parametrize the coset manifold SU(2;1) SU(2)U(1) . The bosonic part of the Lagrangian is e 1 L = 1 2 R 1 2 g XY D q X D q Y P(q) 1 4 F F ; (2.4) where D q X = @ q X + gA K X (q); (2.5) and K X are the four Killing vectors of the gauged isometries on the scalar manifold. Parametrizing the scalar fields byq X =fV; ; R; g, 1 the scalar potential is given by P = g 2 6 3R 2 V + 3R 4 V 2 ; (2.6) and the superpotential is W = 1 + R 2 V : (2.7) The metricg XY of the scalar manifold is ds 2 = 1 2V 2 dV 2 + 1 2V 2 d 2 2R 2 V 2 dd + 2 V dR 2 + 2R 2 V (1 + R 2 V )d 2 : (2.8) ThisN = 2 gauged supergravity with one hypermultiplet in five dimensions can be obtained from the SU(3)-invariant truncation ofN = 8 gauged supergravity in five dimensions. They have identical field content and the scalar manifold. 1 The scalar field,R, was denoted byr in [23]. It should not be confused with the Ricci scalar in (2.4). 19 Now we briefly review the super Janus inN = 2 gauged supergravity in five dimen- sions [23]. The metric is theAdS-domain wall, ds 2 = e 2U(r) ds 2 AdS 4 dr 2 : (2.9) There are also four scalar fields, V = V (r); = (r); R = R(r); = (r); (2.10) which depend on ther-coordinate only. We set the gauge field,A , to vanish. Then by having the supersymmetry variations of fermionic fields,i:e: the spin-3/2 and spin-1/2 fields, to vanish, one obtains the supersymmetry equations, U 0 =gW ; (2.11) V 0 = 6g R 2 + R p V p 1 2 ; (2.12) R 0 = 3g R + R 2 p V p 1 2 ; (2.13) where = s 1 2 e 2U g 2 W 2 ; (2.14) and the scalar fields and are consistently set to be constant. Then, numerically plottingV = V (r), we find that it exhibits the nontrivial profile of the dilaton field in Janus solutions. 20 2.2.3 The Janus solutions in type IIB supergravity Later in [47] the Janus gauge theories were constructed more systematically. It also gives the complete classification of all possible Janus solutions in type IIB supergravity. Isometry SO(6) SU(3) SU(2)U(1) SO(3)SO(3) Supersymmetries zero four eight sixteen Table 2.1 Classification of all possible Janus solutions in type IIB supergravity In table 1 it shows the isometry of internal space and the number of real supersymmetry out of total 32 real supersymmetries of type IIB supergravity. The number of supersym- metry counts real supercharges with both Poincar´ e and conformal supercharges. The Janus solution with no supersymmetry is the original Janus solution. From these obser- vations, Janus solutions with four and sixteen supersymmetries were constructed in [20] and [21, 22] respectively. Here we take a look at the one with four supersymmetries as this one hasSU(3) isometry. We briefly review the supersymmetric Janus solution with the internal space isome- trySU(3) in type IIB supergravity [20]. The metric is given by ds 2 = f 2 4 ds 2 AdS 4 dr 2 + f 2 1 (d + A 1 ) + f 2 2 ds 2 CP 2 ; (2.15) where ds 2 CP 2 = d 2 + 1 4 sin 2 2 1 + 2 2 + cos 2 2 3 ; (2.16) and A 1 = 1 2 sin 2 3 ; (2.17) 21 and i ;i = 1, 2, 3, are theSU(2)-invariant one-forms, 1 = sin 2 cos 3 d 1 + sin 3 d 2 ; 2 = + sin 2 sin 3 d 1 + cos 3 d 2 ; 3 = cos 2 d 1 d 3 ; (2.18) The five-form flux is given by F (5) = f 5 e 0 ^e 1 ^e 2 ^e 3 ^e 4 +e 5 ^e 6 ^e 7 ^e 8 ^e 9 ; (2.19) wheree n ,n = 0; :::; 9 are the frames of the metric, e i = f 4 ^ e i ; e 4 = dr; e 5 = f 1 ^ e 5 = f 1 (d +A 1 ); e a = f 2 ^ e a ; (2.20) wherei = 0, 1, 2, 3,a = 6, 7, 8, 9, and ^ e 6 = d; ^ e 7 = 1 4 sin(2) 3 ; ^ e 8 = 1 2 sin 1 ; ^ e 9 = 1 2 sin 2 : (2.21) The two-form gauge potential is given by B DEG (2) = C (2) iB (2) = if 3 2 ig 3 2 ; (2.22) where B DEG (2) is the two-form gauge potential defined in [20], C (2) and B (2) are RR and NSNS two-form gauge potentials respectively, 2 is the holomorphic (2,0)-form onCP 2 ,f 3 andg 3 are complex functions, and the bar denotes complex conjugation. The dilaton/axion fields are denoted byB with its associated functionf. Overall, the most 22 general solution with the SU(3) isometry of internal space is specified by the seven functions,f 1 ,f 2 ,f 3 ,g 3 ,f 4 ,f 5 ; andB, and they depend only on ther-coordinate. In section 9 of [20], a special case is presented when a DEG = 3 f 1 f 2 2 f (f 3 Bg 3 ) = 0; (2.23) wherea DEG is a function defined for convenience in [20]. Furthermore, in this case, f 1 f 2 = ; f 5 = 3 2f 1 1 2 f 1 f 2 2 ; (2.24) where is a constant. Some functions are integrated to hyper-elliptic integral as f 2 4 @ @r 2 = 1 + C 2 2 9 8 6 2 2 ; (2.25) where = DEG = f 2 f 4 andC 2 is a constant. Here DEG is a quantity defined in [20]. 23 2.3 Truncation ofN = 8 gauged supergravity in five dimensions 2.3.1 TheSU(3)-invariant truncation We study theSU(3)-invariant truncation ofN = 8 gauged supergravity in five dimen- sions. There are a gravitone a , a vector fieldA , and four real scalarsx i for the bosonic field content in theSU(3)-invariant sector. As mentioned in the introduction, there have been studies on theSU(3)-invariant truncation in [14, 6, 7, 17] and [10], but consistently they did not included the dilaton and axion fields in these studies. Here we extend the field content to all four scalar fields including dilaton and axion fields. Let us count the number of bosonic fields in theSU(3)-invariant truncation. In the full theory, under the gauge group,SU(4)'SO(6), 1 gravitone a transforms as1, 15 vector fieldsA IJ as15, 12 two-form tensor fieldsB I as6 +6, and 42 scalar fields abcd as20 0 +10 +10 +1 +1. By breakingSU(4) down toSU(3) they branch as [7] e a 1 ! 1; (2.26) A IJ 15 ! 8 +3 +3 +1; (2.27) B I 6 +6 ! (3 +3) + (3 +3); (2.28) 20 0 ! 8 +6 +6; abcd 10 +10 ! (1 +3 +6) + (1 +3 +6); (2.29) 1 +1 ! 1 +1; so we have a gravitone a , a vector fieldA , and four scalarsx i in theSU(3)-invariant sector. 24 The 42 scalar fields ofN = 8 gauged supergravity in five dimensions live on the coset manifold E 6(6) =USp(8). The basic structure of the coset manifold is explained in [14], and is summarized in appendix A. Fundamental representation ofE 6(6) is real and 27-dimensional. The infinitesimalE 6(6) transformation in theSL(6;R)SL(2;R) basis, (z IJ ,z I ), is [14] z IJ = K I z KJ K J z IK + IJK z K ; (2.30) z I = I K z K + z I + KLI z KL ; (2.31) where I J and are real and traceless generators ofSL(6;R) andSL(2;R) respec- tively, and the coset elements IJK are real and antisymmetric inIJK. Among theE 6(6) generators, theSU(3) generators of the gauge groupSO(6) are the ones that commute with the complex structure, J IJ , which is an antisymmetric tensor with nonzero components,J 12 = J 34 = J 56 = 1. Then we obtain theSU(3)-invariant generators by finding ones that commute with the SU(3) generators. There are eight SU(3)-invariant generators, and they close onto anSU(2; 1) algebra, (1) IJK = + ( 1 3 5 7 IJK 2 4 6 8 IJK ) + ( 1 3 6 8 IJK 2 4 5 7 IJK ) + ( 1 4 5 8 IJK 2 3 6 7 IJK ) ( 1 4 6 7 IJK 2 3 5 8 IJK ); (2.32) (2) IJK = + ( 1 3 5 8 IJK 2 4 6 7 IJK ) + ( 1 3 6 7 IJK + 2 4 5 8 IJK ) + ( 1 4 5 7 IJK + 2 3 6 8 IJK ) ( 1 4 6 8 IJK 2 3 5 7 IJK ); (2.33) (3) IJK = + ( 1 3 5 8 IJK 2 4 6 7 IJK ) + ( 1 3 6 7 IJK 2 4 5 8 IJK ) + ( 1 4 5 7 IJK 2 3 6 8 IJK ) ( 1 4 6 8 IJK 2 3 5 7 IJK ); (2.34) (4) IJK = + ( 1 3 5 7 IJK + 2 4 6 8 IJK ) + ( 1 3 6 8 IJK 2 4 5 7 IJK ) + ( 1 4 5 8 IJK 2 3 6 7 IJK ) ( 1 4 6 7 IJK + 2 3 5 8 IJK ); (2.35) 25 (5)I J = J IJ ; (2.36) (6) = (S 1 ) ; (2.37) (7) = (S 2 ) ; (2.38) (8) = (S 3 ) ; (2.39) where S 1 = 0 @ 0 1 1 0 1 A ; S 2 = 0 @ 1 0 0 1 1 A ; S 3 = 0 @ 0 1 1 0 1 A ; (2.40) are threeSL(2;R) generators. We refer to appendix B for theSU(2; 1) algebra of these generators. The generators (6) , (7) are symmetric, and with the self-duality defined by IJK = + 1 6 IJKLMNP MNP ; (2.41) (1) , (2) are self-dual. By computing the Cartan-Killing form [14] these symmetric and self-dual generators turn out to be the noncompact generators of the scalar manifold [46], M = SU(2; 1) SU(2)U(1) : (2.42) We exponentiate the transformations by four noncompact generators, T 1 = 1 4 p 2 (1) ; T 2 = 1 4 p 2 (2) ; T 3 = 1 2 p 2 ( (7) + (6) ); T 4 = 1 2 p 2 ( (7) (6) ); (2.43) with parameters, x 1 , x 2 , x 3 , x 4 , respectively. Schematically the exponentiation of the generators is z 0 = e (x 3 T 3 +x 4 T 4 ) e (x 1 T 1 +x 2 T 2 ) z: (2.44) 26 From the exponentiation we can extract the coset representatives in the SL(6;R)SL(2;R) basis, U IJ KL , U IJK , U I KL and U I J , by (A.1) and (A.2). The coset representatives in the USp(8) basis,V IJab ,V I ab , are obtained by (A.3) and (A.4). Now with the coset representatives in theUSp(8) basis, we can reduce the bosonic part of the Lagrangian of the SU(3)-invariant truncation. We introduce an angular parametrization of the scalar fields, x 1 = 2 cos ; x 2 = 2 sin ; x 3 = 2 cosa; x 4 = 2 sina: (2.45) The bosonic part of the Lagrangian is e 1 L = 1 4 R +L kin +P 3 4 F F ; (2.46) where the kinetic term for the scalar fields is L kin = 1 2 @ @ + 1 8 sinh 2 (2) @ + sinh 2 @ a + gA 2 + cosh 2 1 2 @ @ + 1 8 sinh 2 (2)@ a@ a ; (2.47) and the scalar potential is P = 3 32 g 2 cosh 2 (2) 4 cosh(2) 5 : (2.48) Note that the scalar potential is manifestly invariant underSL(2;R), i:e: it is indepen- dent of anda. We note that anda are dilaton and axion fields in five dimensions. 27 The scalar potential has two critical points which are theAdS 5 vacua in theSU(3)- invariant truncation [16, 8]. 2 One of the critical points is theN = 8 supersymmetric SO(6) point where = 0 andP = 3 4 g 2 . This point lifts to AdS 5 S 5 vacuum in type IIB supergravity. Another one is the nonsupersymmtric SU(3) point where = 1 2 log(2 p 3) andP = 27 32 g 2 . This point lifts to a solution found by Romans in type IIB supergravity in [49]. The holographic renormalization flows studied in [6, 7, 17, 9] and the domain wall solution for holographic superconductor in [27, 28] flow to this critical point. 2.3.2 The supersymmetry equations In this section we will explicitly derive the supersymmetry equations for the SU(3)- invariant truncation with the dilaton and axion fields, and then solve them numerically. We set the gauge field, A , to vanish. Some equivalent equations inN = 2 gauged supergravity were obtained in [40, 43], however, this subsection is to have equations in the parametrization ofN = 8 gauged supergravity in five dimensions with more scalar fields. We will consider theAdS-domain wall, ds 2 = e 2U(r) ds 2 AdS 4 dr 2 ; (2.49) where ds 2 AdS 4 = 1 z 2 (dt 2 dx 2 dy 2 dz 2 ): (2.50) 2 The scalar field was denoted by' 1 = in [8]. 28 We begin by considering the superpotential and the spinors in five dimensions. The superpotential,W , is obtained as one of the eigenvalues ofW ab tensor [8], W ab b (k) = W a (k) ; (2.51) wherek = 1; 2. There are two eigenvalues with degeneracy of two and six, and they are, respectively, W 1 = 3 4 1 + cosh(2) ; (2.52) W 2 = 1 4 5 + cosh(2) ; (2.53) but onlyW = W 1 gives the scalar potential by P = g 2 8 @W @' i 2 g 2 3 jWj 2 ; (2.54) where' i = ; ; ; a: The eigenvectors, a (1) , a (2) , for the superpotential,W , are a (1) = (0; 1; 0; 1;1; 0; 1; 0); (2.55) a (2) = (1; 0; 1; 0; 0;1; 0;1); (2.56) and they are related to each other by ab b (1) = a (2) ; ab b (2) = + a (1) ; (2.57) 29 where ab is theUSp(8) symplectic form given in e.g. [8]. We employed the gamma matrix conventions in [8]. Then theSU(3)-invariant five-dimensional spinors are given by a = a (1) ^ 1 + a (2) ^ 2 ; (2.58) a = ab b = a (2) ^ 1 + a (1) ^ 2 ; (2.59) where ^ 1 and ^ 2 are spinors with four complex components. The supersymmetry equations are obtained by setting the supersymmetry variations of fermionic fields, i.e. the spin-3/2 and spin-1/2 fields, to zero. For the supersymmetry analysis we will suppress the gauge field, A , below. The purely bosonic parts of the variations are [14] a = D a 1 6 gW ab b ; (2.60) abc = p 2 h P abcd d 1 2 gA dabc d i : (2.61) First we solve the spin-3/2 field variation. For thet-,x-,y- directions, U 0 (4) a e U (3) a 1 3 gW ab b = 0; (2.62) where the prime denotes the derivative with respect to the r-coordinate. We plug the spinors, (2.58), in (2.62) and rearrange to obtain 0 @ U 0 (4) + e U (3) 1 3 gW 1 3 gW U 0 (4) + e U (3) 1 A 0 @ ^ 1 ^ 2 1 A = 0: (2.63) 30 From the integrability of the variation,i:e: the determinant of the matrix in (2.63) van- ishes, we obtain [40, 43] U 0 = 1 3 gW ; (2.64) where = s 1 9e 2U l 2 g 2 W 2 : (2.65) From here the upper and lower signs in equations are related. Note that for the flat domain wall,l!1, we have = 1. By plugging (2.64) back into (2.63), we obtain a projection condition for the spinors, ^ 1 = + ( (4) + p 1 2 (3) ) ^ 2 ; ^ 2 = ( (4) + p 1 2 (3) ) ^ 1 : (2.66) For the flat domain wall limit,l!1, it reduces to the projection condition in [8]. By multiplying (4) on both sides of (2.66) and rearranging them, (4) 0 @ ^ 1 ^ 2 1 A =i 2 4 0 @ 0 i i 0 1 A + p 1 2 (i (4) (3) ) 0 @ 0 1 1 0 1 A 3 5 0 @ ^ 1 ^ 2 1 A : (2.67) This is the first projection condition on the spinors, ^ 1 , ^ 2 . Motivated by the studies on the curved domain wall solutions inN = 2 gauged supergravity in five dimensions [40, 43], we impose another projection condition on the spinors. We define an operator, =i (4) (3) . Noting that 2 = 1, we assume that it acts on the spinors as 0 @ ^ 1 ^ 2 1 A = 0 @ cos sin sin cos 1 A 0 @ ^ 1 ^ 2 1 A : (2.68) 31 where we have introduced a new field, = (r). By solving the supersymmetry equa- tions and the field equations, we will show that it is consistent to impose the projection condition, (2.68), and the new field,(r), is fully determined. Using (2.68) we rewrite (2.67) as (4) ^ i =i h ( 2 ) ij + p 1 2 cos ( 1 ) ij + sin ( 3 ) ij i ^ j ; (2.69) where a ,a = 1; 2; 3, are the Pauli matrices. For brevity, we will write it as (4) ^ i = S ij ^ i ; (2.70) where the components of the matrix,S ij , can be read off from (2.69). Similar projection condition was obtained inN = 2 gauged supergravity in five dimensions in [40, 43]. Now, with the projection condition, (2.69), we solve the spin-1/2 field variation, (2.61), abc = p 2 h P abcd d 1 2 gA dabc d i = 0: (2.71) When we plug the spinors, (2.58), in (2.71), we obtain (P 4abcd d (1) ) (4) ^ 1 + (P 4abcd d (2) ) (4) ^ 2 g 2 (A dabc d (1) ) ^ 1 g 2 (A dabc d (2) ) ^ 2 = 0: (2.72) For any specific choice ofabc indices we define P 1 = P 4abcd d (1) ; P 2 = P 4abcd d (2) ; A 1 = A dabc d (1) ; A 2 = A dabc d (2) : (2.73) 32 It turns out thatA 1 andA 2 can have two distinct values, A d368 d (1) = 3 16 i cos sinh; A d368 d (2) = + 3 16 i sin sinh; (2.74) or A d457 d (1) = 3 16 i sin sinh; A d457 d (2) = 3 16 i cos sinh: (2.75) Similarly, forP 1 andP 2 , we have P 4 368d d (1) = + 1 16 h i sin 0 i sinh cos 0 cosh 2 sina 0 + cosh 2 sinh cosa + i sinh cos sinh 2 2 a 0 ; P 4 368d d (1) = 1 16 h i cos 0 i sinh sin 0 cosh 2 cosa 0 + cosh 2 sinh sina i sinh sin sinh 2 2 a 0 ; (2.76) or P 4 457d d (1) = + 1 16 h i cos 0 i sinh sin 0 cosh 2 cosa 0 + cosh 2 sinh sina i sinh sin sinh 2 2 a 0 ; P 4 457d d (1) = + 1 16 h i sin 0 i sinh cos 0 cosh 2 sina 0 + cosh 2 sinh cosa + i sinh cos sinh 2 2 a 0 : (2.77) 33 Other choice forabc indices gives the same supersymmetry equations in the end. For a particular choice ofP 1 ,P 2 ,A 1 ,A 2 , (2.72) reduces to 2 X i=1 h P i (4) ^ i g 2 A i ^ i i = 0: (2.78) Then plugging the projection condition, (2.70), in (2.78) gives 2 X i;j=1 h P i S ij g 2 A i ij i ^ j = 0: (2.79) Since we want to have the maximal supersymmetry, we assume that the spinors, ^ 1 , ^ 2 , are independent which implies that (2.79) can be solved by having 2 X i=1 h P i S ij g 2 A i ij i = 0; (2.80) wherej = 1, 2. After some calculations, the two complex equations in (2.80) yield four real flow equations, 0 = + 3 2 g p 1 2 cos(a + ) sinh; (2.81) 0 = 3 4 g sinh(2) = g 2 @W @ ; (2.82) a 0 = 3g p 1 2 sin(a + ) csch(2) sinh; (2.83) 0 = + 3 2 g p 1 2 sin(a + ) tanh sinh: (2.84) In appendix C, we show that, with the other choice of P 1 , P 2 , A 1 , A 2 , we lead to the same set of flow equations, so that the spin-1/2 field variation is solved without introducing additional projection condition. We also obtained the field equations and presented them in appendix F. Unlike the flow equations, (2.64) and (2.81)-(2.84), which 34 are first order differential equations, the field equations are second order. A lengthy calculation shows that the flow equations are indeed consistent with the field equations, provided that the field,(r), introduced in (2.68) satisfies 0 = 3 2 g p 1 2 sin(a + ) tanh sinh: (2.85) This first order constraint on the field,(r), is a result of the fact that the supersymmetry equations and the field equations cannot be reduced to a first order system. 3 Also note that the supersymmetry equations imply that in the limit,l!1, which describes a flat domain wall, we must set; a; to be constants, i.e. the dilaton/axion fields decouple, and vice versa. One can turn on the dilaton/axion fields only in the curved domain wall [40, 43]. We have also checked the integrability of the spin-3/2 field variations for ther- and z-directions, but they do not generate any new constraint on the supersymmetry. The variations for these directions are presented in appendix D. By solving the spin-3/2 field variation for ther-direction, @ r ^ 1 (+Q 1 ^ 1 + Q 2 ^ 2 ) + 1 6 gW (4) ^ 2 =0; (2.86) @ r ^ 2 (Q 2 ^ 1 Q 1 ^ 2 ) 1 6 gW (4) ^ 1 =0; (2.87) 3 Here the role of (r) is effectively to reduce one remaining second order equation to a first order equation. 35 where Q 1 =i sinh h cos(a ) 0 1 2 sin(a ) sinh(2)a 0 i ; (2.88) Q 2 =i h sinh sin(a ) 0 i 2 sinh 0 + 1 2 cos(a ) sinh(2) sinh i 2 3 + cosh(2) sinh 2 a 0 i ; (2.89) we obtain ther-dependence of the spinors, 0 @ ^ 1 (r) ^ 2 (r) 1 A = e U=2 0 @ cos 2 sin 2 sin 2 cos 2 1 A 0 @ e i 2 0 0 e i 2 1 A 0 @ ^ (0) 1 ^ (0) 2 1 A ; (2.90) where cos = . Here ^ (0) i , i = 1, 2, depend on theAdS 4 part of the coordinates in (2.49), but are independent of ther-coordinate, and satisfy the projection conditions for the flat domain wall, (4) 0 @ ^ (0) 1 ^ (0) 2 1 A = 0 @ 0 1 1 0 1 A 0 @ ^ (0) 1 ^ (0) 2 1 A ; (2.91) and (i (4) (3) ) 0 @ ^ (0) 1 ^ (0) 2 1 A = 0 @ 1 0 0 1 1 A 0 @ ^ (0) 1 ^ (0) 2 1 A : (2.92) This explains the fact that all the integrability conditions have been satisifed, i:e: an explicit solution to a system of equations must satisfy all integrabilities automatically. Before we close this section, let us count the number of supersymmetries the solution has. Each five-dimensional spinor, ^ i ,i = 1; 2, has four complex components, so we have sixteen real supercharges in total to begin with. The Majorana-Weyl condition 36 on ^ i , i = 1; 2, halves the number of real supercharges to eight. Then, as we have two projection conditions, (2.67) and (2.68), each halves the number of supersymmetry. Hence, there are two real supersymmetries finally. This is the half of the supersymmetry of theSU(3)-invariant flow on the flat domain wall, as the interface of the Janus solution breaks half of the supersymmetry. 2.3.3 The numerical solutions Now we numerically solve the supersymmetry equations, (2.64) and (2.81)-(2.84). We choose the upper sign forr > 0 and the lower sign forr < 0 [23]. From the condition, 0 < < 1, we have 0 < 1 9e 2U l 2 g 2 W 2 < 1; (2.93) where the right hand side is trivially satisfied. From the left hand side, we have 1 3 lgW < e U < + 1 3 lgW: (2.94) As the superpotential, W = 3 4 (1 + cosh(2)), satisfies W > 3 4 , from the left hand side of (2.94), we obtain 1 4 lg < e U : (2.95) From the supersymmetry equations, we have U 00 = 1 l 2 e 2U 3 8 g 2 sinh 2 (2): (2.96) 37 From (2.94), we also have 3e U lg < W < 3e U lg : (2.97) Hence, imposing (2.97) with sinh 2 (2) = 8 3 W + 16 9 W 2 in (2.96), we get a condi- tion, 5 l 2 e 2U + 3g l e U <U 00 < 5 l 2 e 2U + 3g l e U : (2.98) In order to obtain a Janus solution, as it was observed in the previously known Janus solutions, we require the turning point ofU to be a minimum,U 00 > 0. Then, from the right hand side of (2.98), we obtain e U < 3 5 lg: (2.99) From (2.95) and (2.99), there is a narrow range of initial conditions which gives smooth and nonsingular solutions [23], 1 4 lg < e U < 3 5 lg: (2.100) Outside of this range the solution becomes singular at the domain wall i.e. at the origin. A numerical solution in the critical range is plotted in figure 2.1, with the choice of initial conditions, U(0) = 0, (0) = 0:01, (0) = 0:1, (0) = 1, a(0) = 0:1, (0) = 0:1,l = 1, andg = 2. Note that the five-dimensional dilaton and axion fields, and a, exhibit the dilaton profile of Janus solutions, i.e. it takes constant values on both sides of the interface, but jumps across the interface. Indeed we will explicitly identify the solution to be the supersymmetric Janus solution in five dimensions in the next section. 38 -4 -2 2 4 r 0.002 0.004 0.006 0.008 0.010 ¿ -4 -2 2 4 r 1.000 1.005 f -4 -2 2 4 r 0.5 1.0 1.5 2.0 2.5 3.0 U -4 -2 2 4 r 0.0996 0.0998 0.1000 0.1002 0.1004 0.1006 y -4 -2 2 4 r 0.0995 0.1000 0.1005 0.1010 a -4 -2 2 4 r 0.0996 0.0998 0.1000 0.1002 0.1004 0.1006 q Figure 2.1: A numerical solution of the supersymmetry equations 2.4 Super Janus inN = 2 gauged supergravity in five dimensions [23] In section 2.2.2, we reviewed a supersymmetric Janus solution, the super Janus, discov- ered by Clark and Karch inN = 2 gauged supergravity in five dimensions [23]. In this section we will show that the solution in theSU(3)-invariant truncation in the previous section is indeed identical to the super Janus. Now we prove the equivalence of the super Janus and the solution in the SU(3)- invariant truncation. There are four scalar fields living on the scalar manifold, SU(2;1) SU(2)U(1) : fV; ; R; g in the super Janus andf; ; ; ag in the SU(3)-invariant truncation. We can reparametrizefV; ; R; g in terms off; ; ; ag by using the inhomogeneous coordinates, i , i = 1, 2, on the scalar manifold as an intermediate parametrization. We present the details of the reparametrization in appendix E. By employing the reparametrization to the action of theSU(3)-invariant truncation, (2.46), we find that it precisely reduces to the action of the super Janus, (2.4). Then, as the 39 supersymmetry equations, (2.12) and (2.13), are for the special case of constant and , they turn out to be the supersymmetry equations of theSU(3)-invariant truncation, (2.81)-(2.84), with the constant phases, i.e. anda are constant, or more specifically, a + = 0. This proves that the solution of theSU(3)-invariant truncation con- sidered in section 2.3 is indeed equivalent to the super Janus. 2.5 Lift of the SU(3)-invariant truncation to type IIB supergravity We uplift theSU(3)-invariant truncation in section 2.2 to type IIB supergravity by the consistent truncation ansatz. The consistent truncation ans¨ atze for metric and dila- ton/axion fields were presented in [16, 9, 10]. By employing the ansatz, lift of the SU(3)-invariant truncation was performed in [10], however, the five-dimensional dila- ton/axion fields were suppressed. In this section we will lift the five-dimensional dila- ton/axion fields, and as a consequence, we will have nontrivial IIB dilaton/axion fields. We postpone the lift of fluxes to section 2.7. 2.5.1 The metric The ten-dimensional metric is given by ds 2 = 2 ds 2 1;4 + ds 2 5 ; (2.101) 40 where ds 2 1;4 is an arbitrary solution ofN = 8 gauged supergravity in five dimensions. In order to have Janus solution we employ the AdS-domain wall metric, (2.49). The consistent truncation ansatz for the inverse metric of internal space is given by [16, 9, 10] 2 3 g pq = 1 a 2 K IJp K KLq e V IJab e V KLcd ac bd ; (2.102) where e V IJab are the inverse coset representatives of the scalar manifold explained in appendix A,K IJp are Killing vectors on roundS 5 , ab is aUSp(8) symplectic form, = det 1=2 (g mp ^ g pq ), and ^ g pq is the inverse of the roundS 5 metric. The is obtained by taking the determinant on both sides of the ansatz, and 2 = 2 3 is the warp factor. To apply the consistent truncation ansatz, we first prepare the proper coordinates in which theSU(3) isometry of internal space is manifest [10]. In Cartesian coordinates, y I , I = 1, ::: , 6, onR 6 , we think ofS 5 defined by the surface I (y I ) 2 = 1. Let us introduce complex coordinates corresponding to the complex structure,J IJ , u 1 = y 1 + iy 2 ; u 2 = y 5 + iy 6 ; u 3 = y 3 + iy 4 : (2.103) We then introduce the complex coordinates wherez i ,i = 1; 2; are the complex projec- tive coordinates onCP 2 , and' is theU(1) Hopf fiber angle [10], 0 @ u 1 u 2 1 A = u 3 0 @ z 1 z 2 1 A ; u 3 = (1 + z 1 z 1 + z 2 z 2 ) 1=2 e i' : (2.104) Convenient real coordinates for the complex coordinates are [10] 0 @ z 1 z 2 1 A = tan g ( 1 ; 2 ; 3 ) 0 @ 1 0 1 A ; (2.105) 41 where g ( 1 , 2 , 3 ) is anSU(2) invariant matrix in terms of Euler angles, e:g: g ( 1 ; 2 ; 3 ) = 0 @ e i 2 ( 1 + 3 ) cos 2 2 e i 2 ( 1 3 ) sin 2 2 e + i 2 ( 1 3 ) sin 2 2 e + i 2 ( 1 + 3 ) cos 2 2 1 A : (2.106) With the choice of above coordinates, the lifted metric of internal space reduces to ds 2 5 = 1 cosh ds 2 CP 2 + cosh (d' + 1 2 sin 2 3 ) 2 ; (2.107) where ds 2 CP 2 = d 2 + 1 4 sin 2 ( 2 1 + 2 2 + cos 2 2 3 ); (2.108) and i are the left-invariant one-forms ofSU(2), (2.18), which satisfyd i = 1 2 ijk j ^ k . The warp factor in (2.101) is = cosh 1=2 . As mentioned before, lift of the SU(3)-invariant truncation was performed in section 2.9 of [10] without the five- dimensional dilaton/axion fields. Compared to the parametrization of internal space in [10], here we have i ! i ;!;'!'. Besides the parametrization, the lifted metric, (2.107), is identical to the one in [10], i.e. it is independent of the five- dimensional dilaton/axion fields, anda. 2.5.2 The dilaton/axion fields The IIB dilaton/axion fields (; C (0) ) form a complex scalar,, and are related toB by = C (0) + ie = i 1B 1 +B ; (2.109) andf is defined by f = 1 p 1jBj 2 : (2.110) 42 The consistent truncation ansatz for the dilaton/axion fields is given by [9] 4 3 (SS T ) = const V I ab V J cd y I y J ac bd : (2.111) From the ansatz the dilaton/axion field matrix,S, in theSL(2;R) basis reduces to S = 1 2 p 1jBj 2 0 @ 2 + (B +B ) i(BB ) i(BB ) 2 (B +B ) 1 A ; (2.112) where B = ie ia tanh: (2.113) By changing the basis toSU(1; 1), we obtain the dilaton/axion field matrix,V , [9], V = U 1 SU = f 0 @ 1 B B 1 1 A ; U = 0 @ 1 1 i i 1 A ; (2.114) where f = cosh: (2.115) Then from (2.109) the IIB dilaton and axion fields are = ln cosh(2) sin(a) sinh(2) ; (2.116) C (0) = 1 sec(a) coth(2) tan(a) ; (2.117) and we note that they manifestly depend on the five-dimensional dilaton/axion fields, anda. In fugure 2.2 the IIB dilaton and axion fields are plotted with the identical initial condition as figure 2.1. Note that the dilaton and axion fields exhibit the dilaton profile 43 of Janus solutions. Indeed we will explicitly identify our lifted solution as a special case of the supersymmetric Janus solution in type IIB supergravity in the next section. -4 -2 2 4 r 1.215 1.220 1.225 1.230 1.235 1.240 F -4 -2 2 4 r 1.0612 1.0614 1.0616 C 0 Figure 2.2: A numerical solution for the dilaton and axion fields 2.6 Supersymmetric Janus solution in type IIB super- gravity [20] As remarked in the introduction, the supersymmetric Janus solutions in type IIB super- gravity were constructed by D’Hoker, Estes and Gutperle in [20, 21, 22] with variety of supersymmetries and isometries. In this section we will show that by choosing metric and dilaton/axion fields to be the lifted ones in section 2.5, the supersymmetric Janus solution withSU(3) isometry in [20] is completely determined, i.e. this choice fixes all the IIB fields uniquely including three- and five-form fluxes. Now we compare the lifted metric, (2.107), and the dilaton/axion fields, (2.113) and (2.115), in section 2.5 with the supersymmetric Janus solution in type IIB supergravity presented in section 2.2.3. By comparing the metric and the dilaton/axion fields, we find that the metric and dilaton/axion field functions in the supersymmetric Janus solution in type IIB supergravity in section 2.2.3 are given by f 1 = cosh 1=2 ; (2.118) 44 f 2 = cosh 1=2 ; (2.119) f 4 = e U cosh 1=2 ; (2.120) B = ie ia tanh; f = cosh: (2.121) We find that by plugging the above set of functions into the field equations, (6.6), (6.13)- (6.16), and the supersymmetry equations, (7.24)-(7.29), in [20], the remaining functions in the solution are completely determined, and we obtain f 5 = cosh(2) 5 4 cosh 1=2 ; (2.122) f 3 = e i (a ) sinh tanh; (2.123) g 3 =ie i cosh tanh: (2.124) These functions uniquely fix three- and five-form fluxes in (2.19) and (2.22). Further- more, we note that this choice of the functions falls into the special case, a DEG = 0, explained in section 2.2.3, and we obtain the hyper-elliptic integral, f 2 4 @U @r 2 = e 2U + 2 9 C 2 2 e 4U + 1 4 2 9 C 2 2 2 e 10U 1; (2.125) where = e U and = 1. This proves that the lifted metric and the dilaton/axion fields from theSU(3)-invariant truncation in section 2.5 indeed give a special case of the supersymmetric Janus solution in type IIB supergravity in [20]. 45 2.7 Lift of the SU(3)-invariant truncation to type IIB supergravity (continued) In this section we continue the lift of theSU(3)-invariant truncation to type IIB super- gravity, and uplift the three- and five-form fluxes which were not considered in section 2.5. The lift formulae for three- and five-form fluxes were proposed in [18], however, we will find that those formulae do not reproduce the correct fluxes for the curved domain walls. We will propose modified lift formulae for three- and five-form fluxes valid for both the flat and the curved domain walls, and check them for some nontrivial cases including theSU(3)-invariant truncation. 2.7.1 The three-form flux The three-form flux is defined by, e.g. [50, 51], G (3) = dC (2) dB (2) = dC (2) (C (0) + ie )dB (2) = (dC (2) C (0) dB (2) ) ie dB (2) = F (3) ie H (3) ; (2.126) whereC (2) andB (2) are RR and NSNS two-form gauge potentials respectively, and we also define F (3) = dC (2) C (0) dB (2) ; (2.127) H (3) = dB (2) : (2.128) 46 By examining flow solutions in the flat domain walls, a lift formula for the two-form gauge potential was proposed in [18], B pq = kL 2 M (y K V K ab ) V IJab @y I @ p @y J @ q ; (2.129) wherey I are the Cartesian coordinates for anR 6 embedding ofS 5 , p are the intrinsic coordinates on the S 5 ,M = SS T , andS is given in (2.112). However, if we apply the formula to theSU(3)-invariant truncation with dilaton and axion fields, it does not produce the correct two-form gauge potential found in (2.22) with (2.123) and (2.124). What we obtain from (2.129) is a complicted expression and even cannot be expressed in a simple manner by combination of 2 and 2 , as the correct two-form gauge potential is, hence, we do not present the result here. By empirical observation we propose a modified lift formula for two-form gauge potential, B pq = i p 2 4 3 (y K V K ab ) V IJab @y I @ p @y J @ q ; (2.130) where is the warp factor, andB 1 = B (2) ,B 2 = C (2) . We have verified that this lift formula indeed produces the correct two-form gauge potential in section 2.5. There is also another combination of two-form gauge potentials, A (2) = C (2) B (2) = e i tanh cosh + ie ia sinh 2 ; (2.131) where 2 = 1 12 e 3i' sin 2id^ ( 1 + i 2 ) + 1 2 sin(2) ( 1 + i 2 )^ 3 ; (2.132) is the holomorphic (2,0)-form of the internal space [10]. 47 2.7.2 The five-form flux The lift formula for five-form flux was also proposed in [18], however, we will find that it does not reproduce the correct five-form flux for theSU(3)-invariant truncation with dilaton/axion fields in section 2.5. In this subsection we propose a modified lift formula for five-form flux from empirical observations. We consider the metric, ds 2 1;4 = e 2U() ds 2 4 + d 2 ; (2.133) whereds 2 1;4 is any solution ofN = 8 gauged supergravity in five-dimensions, andvol 5 denotes the unit volume form of ds 2 1;4 , and vol 4 of ds 2 4 . We define the geometric W - tensors, f W ab = y I y J cd V Iac V Jbd ; (2.134) f W abcd = + y I y J V Iab V Jcd ; (2.135) and the geometric scalar potential, e P = g 2 32 2W ab f W ab W abcd f W abcd : (2.136) The geometric superpotential, f W , is one of the eigenvalues of f W ab . Before presenting the modified lift formula, let us review the lift formula proposed in [18], F (5) =F +F; (2.137) where F = d ( f Wvol 4 ): (2.138) 48 Applying this formula to the SU(3)-invariant truncation with dilaton and axion fields, with f W = W ,' i = andvol 4 = e 4U in this case, we obtain F = d(Wvol 4 ) = @W @' i @' i @r dr^ vol 4 + 4W @U @r dr^ vol 4 = @W @' i @' i @r + 4W @U @r vol 5 = @W @' i g 2 @W @' i + 4W g 3 W vol 5 = 4 g 8 @W @' i 2 g 3 W 2 ! vol 5 = 4P vol 5 : (2.139) whereP is the scalar potential and is from the supersymmetry equations invoked when taking the derivative of the geometric superpotential. However, it is not the correct five- form flux, (2.19) with (2.122), as the correct one does not have the factor of . 4 Now we propose the modified lift formula for five-form flux, F = 32 g 2 e Pvol 5 + @ f W @ p d p ^ vol 4 ; (2.140) where p are the intrinsic coordinates of internal space. 4 In fact the five-form flux in (2.19) with (2.122), is also the correct five-form flux in [10], which is the lift of theSU(3)-invariant truncation in the flat domain wall i.e. without dilaton and axion fields. Hence, for the flat domain wall, = 1, and the formula produces the correct five-form flux in [10]. 49 By employing the lift formula to the SU(3)-invariant truncation with dilaton and axion fields, we obtain that e P =P, f W = W , so @ f W @ p = 0. Hence, the five-form flux is F (5) = cosh 2 cosh(2) 5 vol 5 cosh(2) 5 2 cosh 2 J 2 ^ J 2 ^ ( + A); (2.141) whereJ 2 is the K¨ ahler form, and is the one-form dual to the Reeb Killing vector to be explained more in section 2.8. This is indeed the five-form flux found in section 2.6. We believe that the modified lift formula, (2.140), generates the correct five-form fluxes for all the flat domain wall cases that the lift formula in [18] was tested. So far we have verified that it does produce the correct five-form flux for the SU(2)U(1)-invariant truncation in section 2.3 of [10] which is more nontrivial case with e P 6=P, f W 6= W and @ f W @ p 6= 0. However, the lift formula only gives the terms of five-form flux which do not involve the gauge field, A , in five dimensions. For the complete five-form flux, we will just present the flux obtained by using the results in [50, 51], F (5) = cosh 2 cosh(2) 5 vol 5 1 2 K^ ( + A)(dA)^ J 2 cosh(2) 5 2 cosh 2 J 2 ^ J 2 ^ ( + A) 1 4 cosh 4 K^ J 2 ^ J 2 dA^ J 2 ^ ( + A); (2.142) where K = sinh 2 (2) @ + sinh 2 @ a + gA dx : (2.143) In this section we proposed the lift formulae for three- and five-form fluxes, (2.130) and (2.140). However, we should stress that we have not derived them from a consis- tent truncation of type IIB supergravity, but have constructed them based on empirical 50 observations. It is possible that some modification would be needed in the general case, as they are modifications of the formulae in [18]. 2.8 Type IIB supergravity on Sasaki-Einstein manifolds [50, 51] Recently there has been notable development in consistent truncation of type IIB super- gravity on Sasaki-Einstein manifolds [50, 51, 52, 53]. In this section, we will show that theSU(3)-invariant truncation ofN = 8 gauged supergravity in five dimensions and its lift to type IIB supergravity in section 2.3, 2.5 and 2.7 provide a particular example of consistent truncation in [50, 51]. Locally the Sasaki-Einstein metric can be written as [50, 51] ds 2 (SE 5 ) = ds 2 (KE 4 ) + ; (2.144) where ds 2 (KE 4 ) is a local K¨ ahler-Einstein metric with positive curvature and is a globally defined one-form dual to the Reeb Killing vector. There are also a globally defined K¨ ahler two-formJ 2 and a (2, 0)-form complex structure 2 , and they satisfy d = 2J 2 ; (2.145) d 2 = 3i^ 2 : (2.146) The type IIB metric is then given by [50, 51] ds 2 = e 2 3 (4U +V ) ds 2 (E) + e 2U ds 2 (KE 4 ) + e 2V ( + A) ( + A); (2.147) 51 where ds 2 (E) is an arbitrary metric on an external five-dimensional spacetime, U and V are scalar functions 5 andA is a one-form defined on the external five-dimensional spacetime. In [50, 51] it was shown that the consistent truncation of type IIB supergravity on Sasaki-Einstein manifolds leads toN = 4 gauged supergravity coupled to two vector mulptiplets in five dimensions. In section 5.3 and 5.4 of [50] and section 3.4.8 of [51], a particular truncation is presented, and for instance, the five-dimensional action for the particular truncation is 6 L kin = 1 2 @ @ 1 8 sinh 2 (2) (@ 1 2 e @ C (0) 3A ) 2 1 8 cosh 2 (@ @ + e 2 @ C (0) @ C (0) ); (2.148) P = + 3 32 g 2 cosh 2 (2) 4 cosh(2) 5 ; (2.149) where and are five-dimensional scalar fields, 7 and andC (0) are dilaton and axion fields of type IIB supergravity respectively. It seems that the axion field is charged under the gauge field in the kinetic term, however, it is only an artifact of this parametrization. The kinetic term in terms of the projective coordinates on the coset manifold, (E.8), shows that theSL(2;R) invariant complex scalar field is not charged under the gauge field. This truncation without the dilaton/axion fields was used to construct a holo- graphic superconductor in [27, 28]. 5 HereU andV have nothing to do with the warp factor,U, in (2.49) and the scalar field,V , in (2.10). 6 In the truncation in section 3.4.8 of [51], the dilaton/axion fields were not considered. 7 Here and have nothing to do with the scalar field,, in (2.10) and the phase, in (2.69). 52 We found that the following reparametrization of the particular truncation precisely reproduces the five-dimensional action, (2.46), and the lifted IIB fields of the SU(3)- invariant truncation in section 2.5 and 2.7, 8 = ; = Tan 1 cos sin(a ) tanh sin cos(a ) tanh ; = ln cosh(2) sin(a) sinh(2) ; C (0) = 1 sec(a) coth(2) tan(a) : (2.150) This proves that the SU(3)-invariant truncation ofN = 8 gauged supergravity and its lift indeed provides a particular example of type IIB supergravity on Sasaki-Einstein manifolds in [50, 51]. 2.9 The SU(2)U(1)-invariant flows with dilaton and axion fields In this chapter, we studied theSU(3)-invariant truncation with dilaton and axion fields, and showed that this truncation and its uplift have theSU(3)-invariant supersymmetric Janus solution as their solutions. On the other hand, as mentioned in section 2.2.2, according to the classification of Janus solutions in type IIB supergravity [47], there are four kinds of solutions with SO(6), SU(3), SU(2)U(1) and SO(3)SO(3) isometries, and each of them has zero, four, eight, and sixteen real supersymmetries, respectively. The one withSU(3)- isometry is what we have constructed in this chapter. However, the supersymmetric 8 We refer to appendix E for this reparametrization 53 Janus solution with isometry ofSU(2)U(1) has not been constructed explicitly so far. Interestingly, there are theSU(2)U(1)-invariant flows withN = 1 [10] andN = 2 [9] supersymmetry in (1.3) and (1.4), respectively. Hence, it is natural to try to include dila- ton and axion fields to theSU(2)U(1)-invariant flow to construct the Janus solution. Now we consider theN = 2 supersymmetic flow [9] with dilaton and axion fields. As we have reviewed in section 1.2, this flow in [9] involves two scalar fields, and, dual to a fermion bilinear and a scalar bilinear, respectively, (1.4). This flow flows to Hades, however, when uplifted to type IIB supergravity, the singularity is resolved. It describes the Coulomb branch ofN = 4 SYM. We have the coset generators, IJK , = 1 12 IJK dx I ^ dx J ^ dx K ^ dy : (2.151) With the complex coordinates,z 1 = x 1 + ix 2 ,z 2 = x 3 ix 4 ,z 3 = x 5 ix 6 , and z 4 = y 1 + iy 2 , we have = 4 X i=1 ' i i + i ; (2.152) where 1 =dz 1 ^ dz 2 ^ dz 3 ^ dz 4 ; 2 =dz 1 ^ dz 2 ^ dz 3 ^ dz 4 ; 3 =dz 1 ^ dz 2 ^ dz 3 ^ dz 4 ; 4 =dz 1 ^ dz 2 ^ dz 3 ^ dz 4 : (2.153) We also consider twoSL(6;R) generators, I J = diag( + ; + ; ; ;2;2); (2.154) 54 with two noncompactSL(2;R) generators which are dilaton and axion fields, (2.37) and (2.38), as before. We take a subtruncation with two scalar fields, and with dilaton and axion fields, anda, 6= 0; = ' 1 = ' 4 6= 0; = 0; ' 2 = ' 3 = 0: (2.155) The scalar fields, and, are dual to the field theory operators in (1.4). In this sector the W ab has two eigenvalues, each with degeneracy of four. One of these two eigenvalues is the superpotential, W = 1 2 1 2 4 cosh(2); (2.156) where = e . This gives the scalar potential by (2.54). The four eigenvectors for the superpotential,W , are a (1) = (0; 0;1; 0; 0; 0; 0; 1); a (2) = (0; 0; 0; 1; 0; 0; 1; 0); a (3) = (1; 0; 0; 0; 0; 1; 0; 0); a (4) = (0;1; 0; 0; 1; 0; 0; 0); (2.157) and they are related to each other by ab b (1) = a (2) ; ab b (2) = + a (1) ; (2.158) ab b (3) = a (4) ; ab b (4) = + a (3) ; (2.159) where ab is theUSp(8) symplectic form given in e.g. [8]. As we have two symplectic pairs of spinors, there are two times more supersymmetry than the flow in section 2.3. Hence, it is anN = 2 supersymmetric RG flow. 55 2.9.1 The supersymmetry variations (I) Now we consider the supersymmetry variations. In order to involve the dilaton and axion fields, anda, as in the previous sections, we employ theAdS-domain wall of (2.49). For simplicity we explicitly assume trivial phase of the dilaton field, i:e: the axion field is trivial,a 0 = 0. Let us consider the integrability conditions of the spin-1/2 field supersymmetry variation, abc = P abcd d g 2 A dabc d ; (2.160) where d are the spinors in (2.58). We choose two different components of the spin-1/2 field, 457 = 0; 368 = 0: (2.161) By iterating these two equations, they reduce to ^ i m ij 4 ^ j = 0; (2.162) where m 11 =ie 4 csch(2) 0 ; m 12 = 2e 4 csch(2) 0 ; m 21 = 2e 4 csch(2) 0 ; m 22 = ie 4 csch(2) 0 : (2.163) If we take other two components of the spin-1/2 field, 457 = 0; 458 = 0; (2.164) 56 we obtain ^ i b m ij 4 ^ j = 0; (2.165) where b m 11 =ie 4 csch(2) 0 + 2 3 ie 10 csch 2 (2) h 2 1e 6 cosh(2) 0 + 3e 6 sinh(2) 0 i ; b m 12 = 2e 4 csch(2) 0 2 3 e 10 csch 2 (2) 1 + e 6 cosh(2) 0 ; b m 21 = 2e 4 csch(2) 0 ; b m 22 = ie 4 csch(2) 0 : (2.166) However, regardless of taking any components, there should be a unique supersymmetry variation. Hence, by comparingm 12 and b m 12 , we conclude that 0 = 0; (2.167) i:e: the dilaton field should be trivial for theN = 2 supersymmetric flow. With 0 = 0 it reduces back to the flow on the flat domain wall in (2.49). Note that we have assumed the axion field,a, to be trivial in this section. 2.9.2 The supersymmetry variations (II) In the previous subsection, we showed that theSU(2)U(1)-invariantN = 2 supersym- metric flow cannot involve nontrivial dilaton field, provided the axion field,a, is trivial. Now, without assuming trivial axion field, we solve the supersymmetry variations, as we have proceeded in section 2.3.2. 57 First we consider the spin-3/2 field variation. For thet-,x-,y- directions, we obtain U 0 (4) a e U (3) a 1 3 gW ab b = 0: (2.168) In the same manner as in section 2.3.2, we obtain the same projection condition, (4) ^ i =i h ( 2 ) ij + p 1 2 cos ( 1 ) ij + sin ( 3 ) ij i ^ j = S ij ^ j ; (2.169) where i ,i = 1; 2; 3, are the Pauli matrices. Now we solve the spin-1/2 field variation in the same manner as in section 2.3.2, abc = p 2 h P abcd d 1 2 gA dabc d i = 0; (2.170) and obtain 2 X i;j=1 h P i S ij g 2 A i ij i ^ j = 0: (2.171) whereS ij is defined in (2.169) and for a specific choice ofabc indices, P 1 = P 4abcd d (1) ; P 2 = P 4abcd d (2) ; A 1 = A dabc d (1) ; A 2 = A dabc d (2) : (2.172) Then we try to solve 2 X i=1 h P i S ij g 2 A i ij i = 0; (2.173) where j = 1, 2. However, unlike the SU(3)-invariant truncation in section 2.3.2, it gives equations involving 0 , a 0 , 0 , 0 , and they contradict with each other. Hence, they cannot be solved for 0 ,a 0 , 0 , 0 . On the other hand, if we set 0 = 0,a 0 = 0, the equations reduce to the flow equations on the flat domain wall in [9]. Therefore, by employing the projection conditions on the spinors, (2.169), it seems that theN = 2 58 supersymmetricSU(2)U(1)-invariant flow does not allow nontrivial dilaton and axion fields. However, there can be additional conditions on the spinors, ^ j , which we have not found yet. It remains as a future work to study if there are any additional projection conditions to solve the supersymmetry variations. We have also considered theN = 1 supersymmetric SU(2)U(1)-invariant flow [10] in (1.4) in the same manner. Like theN = 2 supersymmetic flow here, by employ- ing the projection conditions, (2.169),N = 1 supersymmetric flow seems not allow nontrivial dilaton and axion fields. 59 Chapter 3 Holographic RG flows with gauge fields 3.1 TheSU(3)-invariant flow with electric potentials 3.1.1 The supersymmetry variations Due to the top-down models of AdS/CMT, consistent truncation involving gauge fields has become an interesting topic. Motivated by this, the N = 1 supersymmetric SU(2)U(1)-invariant flow [8, 10] in (1.2) was generalized to involve electric poten- tials, and a flow interpolating between two global AdS 5 was discovered [37]. In this section, in the same spirit of [37], we study the electrically charged SU(3)-invariant flow with and without dilaton and axion fields. 1 We consider the globalAdS background, ds 2 5 = e U(r) f(r) 2 dt 2 d 2 4 ( 2 1 + 2 2 + 2 3 ) dr 2 f(r) 2 ; (3.1) 1 As we have seen in section 2.3.2, there are two superpotentials, (2.52) and (2.53), in the SU(3)- invariant truncation. If we set all scalar fields but one to vanish for the superpotential of theN = 1 supersymmetricSU(2)U(1)-invariant flow in [37], we recover one of the superpotentials of theSU(3)- invariant truncation, (2.53). However, only (2.52) gives the correct scalar potential and flow equations. Hence, the electrically charged flow studied in [37] does not guarantee the existence of the electrically chargedSU(3)-invariant flow, which we are going to consider in this section. 60 whered is a constant parameter, and j are theSU(2) left-invariant one-forms, 1 = cos 3 d 1 + sin 1 sin 3 d 2 ; 2 = sin 3 d 1 sin 1 cos 3 d 2 ; 3 = d 3 + cos 1 d 2 ; (3.2) which satisfyd i = 1 2 ijk j ^ k . The supersymmetry equations are obtained by setting the supersymmetry variations of fermionic fields, i.e. the spin-3/2 and spin-1/2 fields, to zero. The bosonic parts of the variations are [14] a = D a 1 6 gW ab b 1 6 H ab ( + 2 ) b ; (3.3) abc = p 2 P abcd d 1 2 gA dabc d 3 4 H [ab c]j ; (3.4) where D a = @ a + 1 4 ! ij ij a + Q a b b : (3.5) We define H ab = F ab + B ab ; (3.6) where F ab = F IJ V IJab ; B ab = B I V I ab : (3.7) For theSU(3)-invariant truncation, we have F IJ = @ A IJ @ A IJ ; B I = 0; (3.8) 61 and we only consider the solutions with electric charges, A tIJ = (r)J IJ ; (3.9) whereJ IJ is the complex structure, hence, the only non-zero component is H rtab = @ r A tIJ V IJab = 0 J IJ V IJab : (3.10) The eigenvectors, a (1) , a (2) , are related to each other by ab b (1) = a (2) ; ab b (2) = + a (1) ; (3.11) where ab is theUSp(8) symplectic form. Then, as in section 2.3.2, theSU(3)-invariant five-dimensional spinors are defined by a = a (1) ^ 1 + a (2) ^ 2 ; (3.12) a = ab b = a (2) ^ 1 + a (1) ^ 2 ; (3.13) where ^ 1 and ^ 2 are spinors with four complex components. It is convenient to define the quantities,H,Q t , and the superpotential,W , W ab b (1) = W a (1) ; W ab b (2) = W a (2) ; (3.14) H rtab b (1) =H a (2) ; H rtab b (2) = +H a (1) ; (3.15) Q ta b b (1) =Q t a (2) ; Q ta b b (2) = +Q t a (1) ; (3.16) 62 hence, we have W ab b = W a ; (3.17) H rtab b = H rtab ( b (1) ^ 1 + b (2) ^ 2 ) =H a (2) ^ 1 + H a (1) ^ 2 = H a ; (3.18) Q ta b b = Q ta b ( b (2) ^ 1 + b (1) ^ 2 ) =Q t a (1) ^ 1 Q t a (2) ^ 2 =Q t a : (3.19) Then, we further define the quantities, (n) = e U (2H + n d ); e = 2f 1 e U (Q t c d ): (3.20) The time-dependence of the five-dimensional spinors on the globalAdS is given by @ t ^ 1 = c d ^ 2 ; @ t ^ 2 = + c d ^ 1 ; (3.21) wherec is a constant parameter. Now we consider the spin-3/2 field variation, (3.3). For = t; p; r, where p = x; y; z, respectively, the spin-3/2 field variation gives 1 2 (f 0 + fU 0 ) 0 4 a 1 2 e a 1 6 gW 0 b 1 3 (0) 4 a = 0; (3.22) 1 2 fU 0 0 4 a 1 6 gW 0 a + 1 6 (3) 4 a = 0; (3.23) f@ r a + Q ra b b + 1 6 gW 4 b + 1 3 (0) 0 a = 0: (3.24) Now we consider the spin-1/2 field variation, (3.4). It reduces to t P tabcd ( d (1) ^ 1 + d (2) ^ 2 ) + r P rabcd ( d (1) ^ 1 + d (2) ^ 2 ) + g 2 A dabc ( d (1) ^ 1 + d (2) ^ 2 ) 3 2 rt H rtab ( c (2) ^ 1 + c (1) ^ 2 ) = 0; (3.25) 63 With a specific choice ofabc indices we define P tabcd d (1) =P t1 ; P tabcd d (2) =P t2 ; P rabcd d (1) =P r1 ; P rabcd d (2) =P r2 ; (3.26) A dabc d (1) =A 1 ; A dabc d (2) =A 2 ; H rtab c (1) =H 1 ; H rtab c (2) =H 2 : (3.27) Then, we have t (P t1 ^ 1 +P t2 ^ 2 ) + r (P r1 ^ 1 +P r2 ^ 2 ) + g 2 (A 1 ^ 1 +A 2 ^ 2 ) 3 2 rt (H 2 ^ 1 +H 1 ^ 2 ) = 0; (3.28) and it reduces to e U f 1 0 (P t 1 ^ 1 + P t 2 ^ 2 ) + f 4 (P r 1 ^ 1 + P r 2 ^ 2 ) + g 2 (A 1 ^ 1 + A 2 ^ 2 ) 3 2 e U 4 0 (H 2 ^ 1 + H 1 ^ 2 ) = 0: (3.29) 3.1.2 The flow equations without dilaton and axion fields Note that, in theSU(3)-invariant truncation, there are four scalar fields,, ,,a. The scalar field is the phase of, anda is the phase of. Also anda are the dilaton and axion fields, respectively. In this section we first solve the supersymmetry variations only with. From the spin-1/2 field variation, (3.29), we obtain f 0 4 ^ 1 g 2 @W @ ^ 2 + e X 0 ^ 2 = 0; f 0 4 ^ 2 + g 2 @W @ ^ 1 e X 0 ^ 1 = 0: (3.30) 64 Now we collect all the supersymmetry variations, 1 2 (f 0 + fU 0 ) 0 4 a 1 2 e a 1 6 gW 0 b 1 3 (0) 4 a = 0; (3.31) 1 2 fU 0 0 4 a 1 6 gW 0 a + 1 6 (3) 4 a = 0; (3.32) f@ r a + Q ra b b + 1 6 gW 4 b + 1 3 (0) 0 a = 0; (3.33) f 0 4 a g 2 @W @ a + e X 0 a = 0; (3.34) where (n) = e U 3 0 + n d ; e =f 1 e U 3g cosh 2 + 2c d ; e X = 3g 2 e U f 1 sinh(2): (3.35) Subtracting (3.32) from (3.31) gives 1 2 f 0 0 4 a 1 2 e a 1 2 (1) 4 a = 0: (3.36) We consider the dielectric projection condition on the spinors, 0 @ ^ 1 ^ 2 1 A + 0 @ cos 0 sin 4 sin 4 cos 0 1 A 0 @ ^ 1 ^ 2 1 A = 0; (3.37) 65 where = (r). With this projection condition we recast the equations, (3.32), (3.34), (3.36), as cos = f 0 (1) = 1 3 (3) fU 0 = 2 e X g@ W = X f 0 ; (3.38) sin = e (1) = 1 3 gW fU 0 = 2f 0 g@ W = g 12 @ W f 0 : (3.39) From the third equality of (3.38) and the second equality of (3.39), 1 3 (3) (1) fU 0 = 1 g 2 e X @ W + e W ! ; (3.40) and from this, we obtain one of the flow equations, U 0 = g 3c W: (3.41) Then, from the third equality of (3.39), we obtain 0 = gc 2f 2 @W @ : (3.42) From the second equality of (3.39), we obtain 0 + 1 d = 3 g U 0 W g cosh 2 + 2c d ; (3.43) and using this in the second equality of (3.38), ff 0 = 2 3c e 2U g cosh 2 + 2c d : (3.44) 66 One can show that (3.43) and (3.44) are consistent with = 1 2 e U p f 2 c 2 ; (3.45) and d dr h e 3U p f 2 c 2 i = 2 d e 2U : (3.46) Hence, we find the flow equations for the scalar fields, warp factor, and the electric potential similar to the ones in [37]. 3.1.3 The flow equations with dilaton and axion fields In this section, we consider the supersymmetry variations with all four scalar fields,, , , a. The spin-3/2 field variations, (3.31), (3.32), (3.33), do not get modified by including more scalar fields. For the spin-1/2 field variation, (3.29), with all four scalar fields, we obtain for the real part, f 0 4 ^ 1 g 2 1 3 f ( 0 + sinh 2 a 0 ) 4 @W @ ^ 2 + e X 0 ^ 2 cos f 0 4 ^ 2 + g 2 1 3 f ( 0 + sinh 2 a 0 ) 4 @W @ ^ 1 e X 0 ^ 1 sin = 0; (3.47) where e X = g 2 e U f 1 ( 1 + 2 + 3 ) sinh(2): (3.48) For the imaginary part, we obtain 0 (cosa ^ 1 + sina ^ 2 ) + 1 2 sinh(2)a 0 ( sina ^ 1 + cosa ^ 2 ) = 0; (3.49) 67 which gives 0 = 0; (3.50) and = 0 or a 0 = 0: (3.51) Hence, we again conclude that the dilaton field,, should be trivial. 3.2 TheSU(3)-invariant flow with magnetic fields 3.2.1 The magnetic brane solutions Recently, from the AdS/CMT perspective, there were interests in magnetic brane solu- tions in supergravity [31, 32, 33, 34, 35, 36]. In this section, we review the magnetic brane solutions in [32]. We consider the truncation of type IIB supergravity to a five-dimensional Einstein- Maxwell theory [32]. The truncated Lagrangian is L 5 = R 1 4 T 1 ij D T jk T 1 kl D T li 1 8 T 1 ik T 1 jl F ij F kl V ; (3.52) where the scalar potential is V = g 2 2 2T kl T kl (T kk ) 2 ; (3.53) and D i = d i + gA ij j : (3.54) 68 HereT ij is a symmetric 6 6 unimodular tensor to represent the 20 scalar fields in the 20 0 representation of SO(6). The A ij are the one-form potentials to represent the 15 gauge fields. The first ansatz interpolating betweenAdS 5 andAdS 3 T 2 is ds 2 5 =U(r)dt 2 + dr 2 U(r) + e 2V (r) (dx 1 ) 2 + (dx 2 ) 2 + e 2W (r) dy 2 ; (3.55) with T ij = 0 B B B B B B B B B B B B B @ T 1 0 0 0 0 0 0 T 1 0 0 0 0 0 0 T 2 0 0 0 0 0 0 T 2 0 0 0 0 0 0 T 3 0 0 0 0 0 0 T 3 1 C C C C C C C C C C C C C A ; (3.56) F ij (2) = 0 B B B B B B B B B B B B B @ 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 2 0 0 0 0 2 0 0 0 0 0 0 0 0 3 0 0 0 0 3 0 1 C C C C C C C C C C C C C A F (2) ; (3.57) and F (2) =Bdx 1 ^ dx 2 : (3.58) By solving the equations of motion, we obtain ds 2 5 = r 2 L 2 dt 2 + (dx 1 ) 2 + (dx 2 ) 2 + dy 2 + L 2 r 2 dr 2 ; (3.59) 69 which is the expected ultraviolet solution atr !1 whereB = 0. At infrared where B6= 0, we obtain ds 2 5 = 3 (r 2 r 2 + ) L 2 dt 2 + L 2 dr 2 3 (r 2 r 2 + ) + BL p 3 (dx 1 ) 2 + (dx 2 ) 2 + 3r 2 L 2 dy 2 ; (3.60) where L 2 = g 2 3 1 T 1 + 1 T 2 + 1 T 3 ; B 2 = 1 4 2 1 T 2 1 + 2 2 T 2 2 + 2 3 T 2 3 B 2 : (3.61) This is the product of a BTZ black hole and a torus. At zero temperature limit, we obtain AdS 3 T 2 . The second ansatz interpolating betweenAdS 5 andAdS 2 T 3 is ds 2 5 =U(r)dt 2 + dr 2 U(r) + e 2V (r) (dx 1 ) 2 + (dx 2 ) 2 + (dx 3 ) 2 ; (3.62) withT ij identical to the previous one and F ij (2) = 0 B B B B B B B B B B B B B @ 0 1 F 1 (2) 0 0 0 0 1 F 1 (2) 0 0 0 0 0 0 0 0 2 F 2 (2) 0 0 0 0 2 F 2 (2) 0 0 0 0 0 0 0 0 3 F 3 (2) 0 0 0 0 3 F 3 (2) 0 1 C C C C C C C C C C C C C A ; (3.63) 70 with F 1 (2) =B 1 dx 2 ^ dx 3 ; F 2 (2) =B 2 dx 3 ^ dx 1 ; F 3 (2) =B 3 dx 1 ^ dx 2 : (3.64) By solving the equations of motion, we obtain ds 2 5 = g 2 r 2 dt 2 + (dx 1 ) 2 + (dx 2 ) 2 + (dx 3 ) 2 + dr 2 g 2 r 2 ; (3.65) which is the expected ultraviolet solution atr !1 whereB = 0. At infrared where B6= 0, we obtain ds 2 5 = 8g 2 (r 2 r 2 + )dt 2 + dr 2 8g 2 (r 2 r 2 + ) + B g p 2 (dx 1 ) 2 + (dx 2 ) 2 + (dx 3 ) 2 ; (3.66) where T 1 = T 2 = T 3 = 1 and L 1 = g. At zero temperature limit, we obtain AdS 2 T 3 . 3.2.2 Configurations of magnetic fields Motivated by the magnetic brane solutions in the previous section, we study the holo- graphic RG flows in the presence of magnetic fields. To be specific we study theSU(3)- invariant truncation with magnetic fields. Instead of the global AdS, we consider the Poincar´ eAdS background, ds 2 5 = e U(r) f(r) 2 dt 2 dx 2 dy 2 dz 2 dr 2 f(r) 2 : (3.67) 71 In theSU(3)-invariant truncation there is only one gauge field, and we have two choices, A x = A 1 (r); A y = A 2 (r); A z = A 3 (r); (3.68) or A x = A 1 (x;y;z); A y = A 2 (x;y;z); A z = A 3 (x;y;z): (3.69) We will consider the first and second cases in section 3.2.3 and section 3.2.4, respec- tively. 3.2.3 TheSU(3)-invariant flow with magnetic fields (case I) We consider the background in (3.67). We define H ab = F ab + B ab ; (3.70) where F ab = F IJ V IJab ; B ab = B I V I ab : (3.71) For theSU(3)-invariant truncation, we have F IJ = (@ A @ A )J IJ ; B I = 0; (3.72) whereJ IJ is the complex structure. We consider the magnetic components of the gauge field, A x = A 1 (r); A y = A 2 (r); A z = A 3 (r): (3.73) 72 Hence, the non-zero components are H rxab =@ r A x J IJ V IJab ; H ryab =@ r A y J IJ V IJab ; H rzab =@ r A z J IJ V IJab : (3.74) Now we solve the supersymmetry variations. From ta = 0, from xa = 0, ya = 0, za = 0, and from abc = 0, we obtain, respectively, 1 6 gW 4 a 1 2 (f 0 +fU 0 ) a + 1 3 e U f (H 1 1 +H 2 2 +H 3 3 ) a = 0; (3.75) 1 2 fU 0 4 a + 1 6 gW a + 1 3 e U (Q 1 1 + Q 2 2 + Q 3 3 ) a = 0; (3.76) f 0 4 a + g 2 @W @ a + (X 1 1 + X 2 2 +X 3 3 ) a = 0; (3.77) where H i = 3 2 A 0 i ; Q i = gWA i ; X i = g 2 @W @ 2e U A i : (3.78) Now we consider a projection condition on the spinors, a sin 4 a + c 1 1 + c 2 2 + c 3 3 a = 0; (3.79) where sin 2 + c 2 1 + c 2 2 + c 2 3 = 1; (3.80) or c 2 1 + c 2 2 + c 2 3 = cos 2 : (3.81) 73 We recast the equations, (3.75), (3.76), (3.77), as sin = fU 0 1 3 gW = 1 3 gW f 0 + fU 0 = f 0 g 2 @W @ ; (3.82) c i = 2e U Q i gW = 2 3 e U fH i f 0 + fU 0 = X i g 2 @W @ ; (3.83) or after plugging (3.78), sin = fU 0 1 3 gW = 1 3 gW f 0 + fU 0 = f 0 g 2 @W @ ; (3.84) c i = 2e U A i = e U fA 0 i f 0 + fU 0 = 2e U A i : (3.85) We do not present the procedure, but just the first order system they reduce to, U 0 = 1 3 gWh; (3.86) 0 = g 2 @W @ h; (3.87) f 0 = 1 3 gW 1 fh fh ; (3.88) A 0 i = 2A i U 0 1 f 2 h 2 ; (3.89) where the new function, h = h(r), can be determined from the condition, (3.80) or (3.81), as 4e 2U (A 2 1 + A 2 2 + A 2 3 ) = 1 f 2 h 2 : (3.90) From (3.85), we haveA 1 = A 2 = A 3 , hence, (3.90) gives A i = 1 2 p 3 e U p 1 f 2 h 2 : (3.91) 74 By differentiating (3.91) and using the flow equations and the conditions known so far, we obtain A 0 i = A i U 0 f h @h @f ; (3.92) and it should be identical to (3.89), hence, we obtain f h @h @f = 2 f 2 h 2 : (3.93) By integrating (3.93), we obtain h = r 3 2 f 2 ; (3.94) where we have seth = 1 whenf = 1. Hence, we obtain the flow equations, U 0 = 1 3 gW r 3 2 f 2 ; (3.95) 0 = g 2 @W @ r 3 2 f 2 ; (3.96) f 0 = 1 3 gW 0 @ 1 f q 3 2 f 2 f r 3 2 f 2 1 A ; (3.97) A i = 1 2 e U p 1 f 2 : (3.98) They satisfy the projection condition as they reduce to sin = p 3f 2 2; c i = 2e U A i = p 1 f 2 ; (3.99) and sin 2 + c 2 1 + c 2 2 + c 2 3 = (3f 2 2) + 3 (1 f 2 ) = 1: (3.100) Whenf = 1, it reduces back to the flat domain wall solution with no magnetic fields. 75 However, we have also solved the field equations for this truncation, and the flow equations we obtained here were not consistent with the field equations. It remains as a future work to find out why the supersymmetry equations were not sufficient to solve the field equations for this truncation with magnetic fields. 3.2.4 TheSU(3)-invariant flow with magnetic fields (case II) We consider the background in (3.67) with the magnetic field, A x = A x (x;y;z); A y = A y (x;y;z); A z = A z (x;y;z): (3.101) Hence, the non-zero components ofH ab are H xyab = (@ x A y @ y A x )J IJ V IJab ; H yzab = (@ y A z @ z A y )J IJ V IJab ; H zxab = (@ z A x @ x A z )J IJ V IJab : (3.102) Now we solve the spin-3/2 field supersymmetry variation. From ta = 0, we have 1 2 (f 0 + fU 0 ) 4 a 1 6 gW a 1 2 e 2U (H 3 1 2 + H 1 2 3 + H 2 3 1 ) a = 0; (3.103) and from xa = 0, ya = 0, za = 0, we have 1 2 fU 0 4 a 1 3 e U (Q 1 1 + Q 2 2 + Q 3 3 ) a 1 6 gW a + 1 2 e 2U (H 3 1 2 + H 1 2 3 + H 2 3 1 ) a = 0; (3.104) 76 where H 1 = @ x A y @ y A x ; H 2 = @ y A z @ z A y ; H 3 = @ z A x @ x A z ; (3.105) and Q 1 = gWA x ; Q 2 = gWA y ; Q 3 = gWA z : (3.106) Hence, we have two independent equations from the spin-3/2 field variation, (3.103) and (3.104), and they do not reduce to a unique projection condition. This implies that we have two projection conditions in this case. Solving for the supersymmetry equations remains as a future work. 77 Chapter 4 Conclusions We have considered the generalizations of holographic renormalization group flows. First, we studied theSU(3)-invariant truncation ofN = 8 gauged supergravity in five dimensions with dilaton and axion fields and its lift to type IIB supergravity [11]. We showed that the known Janus solutions in five and in ten dimensions, i.e. the super Janus in five dimensions [23] and the supersymmetric Janus solution withSU(3) isometry in type IIB supergravity [20], are constructed in a unified way in the framework ofN = 8 gauged supergravity and its lift. Furthermore, we studied theSU(2)U(1)-invariantN = 1 andN = 2 supersymmetric RG flows with dilaton and axion fields. Second, we studied the SU(3)-invariant RG flow with gauge fields. We found the systems of first-order flow equations for the SU(3)-invariant flows with electric potentials or magnetic fields. As a future work, it would be interesting to find some charged black hole solutions of these systems. Furthermore, it is natural to revisit the SU(2)U(1)-invariantN = 1 supersymmetric RG flow in [37] with magnetic fields, instead of electric potentials, and find some magnetically charged black hole solutions of this truncation. 78 Appendix A:N = 8 gauged supergravity in five dimensions In this appendix we reviewN = 8 gauged supergravity in five dimensions with emphasis on the structure of its scalar manifold,E 6(6) =USp(8), by following [14]. We will employ the conventions of [14] throughout the paper. The SO(6) gaugedN = 8 supergravity in five dimensions [12, 13, 14] has local USp(8) symmetry, but global E 6(6) symmetry of the ungauged theory is broken. The field content consists of 1 gravitone a , 8 gravitini a , 15 vector fieldsA IJ , 12 two- form tensor fields B I , 48 spinor fields abc , and 42 scalar fields abcd where a, b, ::: are USp(8) indices, I, J, ::: are SL(6;R), and , , ::: are SL(2;R). Here SL(6;R)SL(2;R) is one of the maximal subgroups ofE 6(6) . The infinitesimalE 6(6) transformation in theSL(6;R)SL(2;R) basis, (z IJ ,z I ), in terms of I J , , and IJK was already given in (2.30) and (2.31). Exponentiating the transformation in (2.30) and (2.31), z 0 IJ = 1 2 U MN IJ z MN + r 1 2 U PIJ z P ; (A.1) z 0K = U P K z P + r 1 2 U IJK z IJ ; (A.2) we obtain the coset representatives in the SL(6;R)SL(2;R) basis, U IJ KL , U IJK andU I J . We also have the coset representatives in theUSp(8) basis, V IJab = 1 8 ( KL ) ab U IJ KL + 2( K ) ab U IJK ; (A.3) V I ab = 1 4 r 1 2 ( KL ) ab U I KL + 2( K ) ab U I K : (A.4) 79 The inverse coset representatives are e V IJab = 1 8 [( KL ) ab e U IJ KL + 2 ( K ) ab e U IJ K ]; (A.5) e V I ab = 1 4 r 1 2 [( KL ) ab e U IKL + 2 ( K ) ab e U I K ]: (A.6) Now we consider the action of the theory [14]. The bosonic part of the Lagrangian is e 1 L = 1 4 R +L kin +P 1 8 H ab H ab + 1 8ge B I D B I +L CS ; (A.7) where the covariant derivative is defined by D X aI = @ X aI + Q a b X bI gA IJ X aJ ; (A.8) with theUSp(8) connection, Q a b = 1 3 h e V bcIJ @ V IJac + e V bcI @ V Iac +gA IL JL (2V ae IK e V be JK V Jae e V beI ) i : (A.9) The kinetic term for scalar fields is defined by L kin = 1 24 P abcd P abcd ; (A.10) where P abcd = e V ab IJ D V IJcd + e V abI D V I cd : (A.11) 80 The scalar potential is defined by P = 1 32 (2W ab W ab W abcd W abcd ); (A.12) where W abcd = IJ V Iab V Jcd ; (A.13) W ab = W c acb : (A.14) We also define H ab = F ab +B ab ; (A.15) where F ab = F IJ V IJab ; (A.16) B ab = B I V I ab ; (A.17) for the last three terms of Lagrangian. We adopt the gamma matrix convention of [14], with f i ; j g = 2 ij ; (A.18) where ij = diag (+;;;;), and 0 ; 1 ; 2 ; 3 are pure imaginary as in four- dimensions and 4 = i 5 is pure real. The matrices 0 and 5 are antisymmetric and 1 ; 2 ; 3 are symmetric. Only in section 2, in order to prevent the confusion with the function, , we denote the gamma matrices by (i) instead of i . 81 Appendix B:SU(2;1) algebra TheSU(2; 1) algebra is given by [L i ; L j ] = if ijk L k ; (B.1) with the structure constants f 123 = 1; f 147 =f 165 =f 246 =f 257 =f 345 =f 376 = 1 2 ; f 458 =f 678 = p 3 2 : (B.2) The standard 3-dimensionalSU(2; 1) generators are obtained by modifyingSU(3) Gell- Mann matrices where the Gell-Mann matrices are 1 = 0 B B B @ 0 1 0 1 0 0 0 0 0 1 C C C A ; 2 = 0 B B B @ 0 i 0 i 0 0 0 0 0 1 C C C A ; 3 = 0 B B B @ 1 0 0 0 1 0 0 0 0 1 C C C A ; 4 = 0 B B B @ 0 0 1 0 0 0 1 0 0 1 C C C A ; 5 = 0 B B B @ 0 0 i 0 0 0 i 0 0 1 C C C A ; 6 = 0 B B B @ 0 0 0 0 0 1 0 1 0 1 C C C A ; 7 = 0 B B B @ 0 0 0 0 0 i 0 i 0 1 C C C A ; 8 = 1 p 3 0 B B B @ 1 0 0 0 1 0 0 0 2 1 C C C A : (B.3) Multiplying four of the Gell-Mann matrices byi, they close onto anSU(2; 1) algebra, L 1 = 1 2 ; L 2 = 2 2 ; L 3 = 3 2 ; L 4 = i 4 2 ; L 5 = i 5 2 ; L 6 = i 6 2 ; L 7 = i 7 2 ; L 8 = 8 2 ; (B.4) 82 whereL 1 ,L 2 ,L 3 areSU(2) generators,L 4 ,L 5 ,L 6 ,L 7 are SU(2;1) SU(2)U(1) coset generators, andL 8 is aU(1) generator. The generators in the 27-dimensional representation in section 2.2 corresponding to the 3-dimensional generators are given by L 1 ! i 8 ( (3) (4) ); L 2 ! i 8 ( (3) + (4) ); L 3 ! i 4 ( (5) (8) ); L 4 ! i 4 p 2 (1) ; L 5 ! i 4 p 2 (2) ; L 6 ! i 2 p 2 ( (7) + (6) ); L 7 ! i 2 p 2 ( (7) (6) ); L 8 ! i 4 p 3 ( (5) +3 (8) ): (B.5) 83 Appendix C: The supersymmetry variations for spin-1/2 fields In this appendix we reconsider the spin-1/2 field variation, and show that it is solved without introducing additional projection condition. As in (2.73), for a choice of abc indices we defineP 1 ,P 2 ,A 1 ,A 2 . In the same manner, for another choice ofabc indices we define e P 1 , e P 2 , e A 1 , e A 2 . From the spin-1/2 field variation, as in (??), they satisfy P 1 (4) ^ 1 + P 2 (4) ^ 2 g 2 A 1 ^ 1 g 2 A 2 ^ 2 = 0; (C.1) e P 1 (4) ^ 1 + e P 2 (4) ^ 2 g 2 e A 1 ^ 1 g 2 e A 2 ^ 2 = 0: (C.2) By multiplying byA i , e A i and subtracting, we obtain (P 1 e A 2 e P 1 A 2 ) (4) ^ 1 + (P 2 e A 2 e P 2 A 2 ) (4) ^ 2 g 2 (A 1 e A 2 e A 1 A 2 ) ^ 1 = 0; (C.3) (P 1 e A 1 e P 1 A 1 ) (4) ^ 1 + (P 2 e A 1 e P 2 A 1 ) (4) ^ 2 g 2 (A 2 e A 1 e A 2 A 1 ) ^ 2 = 0: (C.4) By rearranging, we obtain ^ 1 2 g P 1 e A 2 e P 1 A 2 A 1 e A 2 e A 1 A 2 (4) ^ 1 2 g P 2 e A 2 e P 2 A 2 A 1 e A 2 e A 1 A 2 (4) ^ 2 = 0; (C.5) ^ 2 2 g P 1 e A 1 e P 1 A 1 A 2 e A 1 e A 2 A 1 (4) ^ 1 2 g P 2 e A 1 e P 2 A 1 A 2 e A 1 e A 2 A 1 (4) ^ 2 = 0: (C.6) 84 Hence, the spin-1/2 field variation, (2.61), reduces to ^ i m ij (4) ^ j = 0; (C.7) where m ij = 0 @ m 1 + m 2 m 3 m 4 m 5 m 4 + m 5 m 1 + m 2 m 3 1 A ; (C.8) and m 1 = 2 3g i csch sin(a ) 0 + 1 2 sinh(2) cos(a )a 0 ; m 2 = 2 3g sinh 2 a 0 ; m 3 = 2 3g 0 ; m 4 = + 2 3g i csch cos(a ) 0 + 1 2 sinh(2) sin(a )a 0 ; m 5 = + 4 3g csch(2) 0 : (C.9) When we plug the supersymmetry equations, (2.81), (2.82), (2.83), (2.84), in (C.7), it reduces to the projection condition, (2.69). Hence, the spin-1/2 field variation, (C.7), does not provide any additional projection condition. In total, we have two projection conditions, (2.67) and (2.68), on the spinors. 85 Appendix D: The supersymmetry variations for spin-3/2 fields In this appendix we present theSU(3)-invariant truncation of supersymmetry variations for spin-3/2 fields. The variation for t-, x-, y- directions is given in (2.62). For z- direction the variation is given by 2e U (3) z@ z ^ 1 U 0 (4) ^ 1 + 1 3 gW ^ 2 = 0; (D.1) +2e U (3) z@ z ^ 2 +U 0 (4) ^ 2 + 1 3 gW ^ 1 = 0: (D.2) For the variation in the r-direction we need to know the action of Q a b tensor on the spinors, Q ra b (1)b = +Q 1 (1)a + Q 2 (2)a ; (D.3) Q ra b (2)b =Q 2 (1)a Q 1 (2)a ; (D.4) where Q 1 =i sinh h cos(a ) 0 1 2 sin(a ) sinh(2)a 0 i ; (D.5) Q 2 =i h sinh sin(a ) 0 i 2 sinh 0 + 1 2 cos(a ) sinh(2) sinh i 2 3 + cosh(2) sinh 2 a 0 i : (D.6) 86 Then the variation in ther-direction is given by @ r ^ 1 (+Q 1 ^ 1 + Q 2 ^ 2 ) + 1 6 gW (4) ^ 2 = 0; (D.7) @ r ^ 2 (Q 2 ^ 1 Q 1 ^ 2 ) 1 6 gW (4) ^ 1 = 0; (D.8) whereW is the superpotential in (2.52). 87 Appendix E: The parametrizations of the scalar manifold In this paper we have employed several different parametrizations for the four real scalar fields living on the scalar manifold, SU(2;1) SU(2)U(1) . In this appendix we summarize the origins of and the relations between different parametrizations. The coset manifold, SU(2;1) SU(2)U(1) , is topologically an open ball inC 2 with the Bergman metric [54], ds 2 = d 1 d 1 + d 2 d 2 1 1 1 2 2 + ( 1 d 2 + 2 d 2 )( 1 d 2 + 2 d 2 ) (1 1 1 2 2 ) 2 ; (E.1) which is a K¨ ahler metric with K¨ ahler potential, K = 1 2 ln(1 1 1 2 2 ): (E.2) The first two parametrizations of the scalar manifold we employed in this paper were the rectangular and angular parametrizations, fx 1 ; x 2 ; x 3 ; x 4 g in (2.44) and f; ; ; ag, respectively, for the the SU(3)-invariant truncation in section 2.2. The relation between them is given in (2.45). In terms of the rectangular parametrization, the inhomogeneous coordinates on the scalar manifold are given by 1 = (x 1 + ix 2 ) tanh 1 2 p x 2 1 + x 2 2 p x 2 1 + x 2 2 sech 1 2 q x 2 3 +x 2 4 ; (E.3) 2 = (x 3 + ix 4 ) tanh 1 2 p x 2 3 + x 2 4 p x 2 3 + x 2 4 : (E.4) 88 We can reverse the relation to get x 1 = 1 + 1 2Z 1 ; x 2 = 1 1 2i Z 1 ; x 3 = 2 + 2 2Z 2 ; x 4 = 2 2 2i Z 2 ; (E.5) where Z 1 = q 1 1 q 1 + 2 2 2 tanh 1 p 1 1 ; Z 2 = q 2 2 2 tanh 1 p 2 2 : (E.6) Before proceeding to the third parametrization, we consider the SU(3)-invariant truncation in terms of the complex coordinates, i ,i = 1, 2. When we exponentiate the coset generators in (2.44), if we employ the complex coordinates by (E.5), we can have the action of theSU(3)-invariant truncation in terms of the complex coordinates, e 1 L = 1 4 R +L kin +P 3 4 F F +L CS : (E.7) The kinetic term is L kin = 1 2 g ij D i D j ; (E.8) where the metric is the Bergman metric, (E.1), and the covariant derivative with respect to the gauge field is D 1 = @ 1 + 3gA 1 ; D 2 = @ 2 : (E.9) The scalar potential is P = 3 8 g 2 (1j 2 j 2 ) (2 3j 1 j 2 2j 2 j 2 ) (1j 1 j 2 j 2 j 2 ) 2 : (E.10) Thirdly, in N = 2 gauged supergravity in five dimensions, there is another parametrization by the scalar fields,fV; ; R; g, which was employed for the super 89 Janus in section 2.3. In terms of these scalar fields the inhomogeneous coordinates are given by [46, 23] 1 = 2iRe i 1 + R 2 + V + i ; (E.11) 2 = 1 R 2 V i 1 + R 2 + V + i : (E.12) By plugging (E.11), (E.12) into (E.7), we precisely reproduce the action of the super Janus, (2.4). The rest of the truncation, e.g. the supersymmetry equations, can also be reparametrized, and they are explained in section 2.3. This reparametrization was used to establish the equivalence of the SU(3)-invariant truncation and the super Janus in section 2.3. Lastly, there is a parametrization byf; ; ; C (0) g, employed for a particular trun- cation of type IIB supergravity on Sasaki-Einstein manifolds in section 2.7. The and C (0) are the IIB dilaton and axion fields respectively, and and are some five- dimensional scalar fields. We briefly mention that by comparing Killing vectors for (2.47) and (2.148), we have found the relation betweenf; ; ; C (0) g andf; ; ; ag in (2.150). Note that the IIB dilaton and axion fields are indeed identical to the ones from the lift in (2.116) and (2.117). 90 Appendix F: The field equations of the SU(3)-invariant truncation In this appendix, we present the field equations of theSU(3)-invariant truncation. Let us consider the action for complex scalar fields and gravity, L = p g 1 4 R + 1 2 g h ab @ a @ b P( a ; a ) : (F.1) The scalar equations reduce to 1 p g @ ( p gg @ a ) + a bc g @ b @ c h ba @ b P = 0; (F.2) 1 p g @ ( p gg @ a ) + a bc g @ b @ c h ab @ b P = 0; (F.3) where a bc = h da @ c h bd ; (F.4) a b c = h ad @ c h db : (F.5) The Einstein equations are R 1 2 Rg = 2T ; (F.6) where the energy-momentum tensor is T = h ab @ a @ b g 1 2 g h ab @ a @ b P(;) : (F.7) 91 For the metric in (2.49), the Einstein equations reduce to 3 (U 00 + 2U 0 U 0 ) + 3 l 2 e 2U = 2 ( 1 2 h ab 0 a 0 b P); (F.8) 3U 0 U 0 + 3 l 2 e 2U = ( 1 2 h ab 0 a 0 b +P): (F.9) Then, for the SU(3)-invariant truncation in (E.7), in terms of the inhomogeneous coordinates,f 1 ; 2 g, the field equations reduce to 0 = 4U 0 0 1 + 00 1 + 2 0 1 ( 1 0 1 + 2 0 2 ) 1 1 1 2 2 + 3g 2 4 1 3 1 1 2 2 2 1 1 1 2 2 ; (F.10) 0 = 4U 0 0 2 + 00 2 + 2 0 2 ( 1 0 1 + 2 0 2 ) 1 1 1 2 2 ; (F.11) 0 = 3 (U 00 + 2U 0 U 0 ) + 3 l 2 e 2U + 2 1 2 0 1 0 1 + 0 2 0 2 1 1 1 2 2 + 1 2 ( 1 0 1 + 2 0 2 )( 0 1 1 + 0 2 2 ) (1 1 1 2 2 ) 2 3g 2 8 (1 2 2 )(2 3 1 1 2 2 2 ) (1 1 1 2 2 ) 2 ; (F.12) 0 = 3U 0 U 0 + 3 l 2 e 2U 1 2 0 1 0 1 + 0 2 0 2 1 1 1 2 2 + 1 2 ( 1 0 1 + 2 0 2 )( 0 1 1 + 0 2 2 ) (1 1 1 2 2 ) 2 3g 2 8 (1 2 2 )(2 3 1 1 2 2 2 ) (1 1 1 2 2 ) 2 : (F.13) 92 Bibliography [1] J. 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Abstract (if available)

Abstract The AdS/CFT correspondence conjectures the duality between type IIB supergravity on AdS_5 x S^5 and N = 4 super Yang-Mills theory. Mass deformations of N = 4 super Yang-Mills theory drive renormalization group (RG) flows. Holographic RG flows are described by domain wall solutions interpolating between AdS_5 geometries at critical points of N = 8 gauged supergravity in five dimensions. In this thesis we study two directions of generalizations of holographic RG flows. ❧ First, motivated by the Janus solutions, we study holographic RG flows with dilaton and axion fields. To be specific, we consider the SU(3)-invariant flow with dilaton and axion fields, and discover the known supersymmetric Janus solution in five dimensions. Then, by employing the lift ansatz, we uplift the supersymmetric Janus solution of the SU(3)-invariant truncation with dilaton and axion fields to a solution of type IIB supergravity. We identify the uplifted solution to be one of the known supersymmetric Janus solution in type IIB supergravity. Furthermore, we consider the SU(2)xU(1)-invariant N = 2 and N = 1 supersymmetric flows with dilaton and axion fields. ❧ Second, motivated by the development in AdS/CMT, we study holographic RG flows with gauge fields. We consider the SU(3)-invariant flow with electric potentials or magnetic fields, and find first-order systems of flow equations for each case.

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Creator Suh, Minwoo (author)

Core Title Generalizations of holographic renormalization group flows

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Publication Date 07/15/2012

Defense Date 03/21/2012

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

Tag OAI-PMH Harvest,string theory,supergravity,the AdS/CFT correspondence

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Pilch, Krzysztof (committee chair), Bars, Itzhak (committee member), Malikov, Fedor (committee member), Shakeshaft, Robin (committee member), Warner, Nicholas P. (committee member)

Creator Email minsuh@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-57103

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