Supersymmetric solutions and black hole microstate geometries

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Content Supersymmetric solutions and black hole microstate geometries by Alexander Tyukov A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS) August 2018 Copyright 2018 Alexander Tyukov Acknowledgments First and foremost I would like express my sincere gratitude to my PhD advisor Nick Warner for his guidance, support and encouragement over the last five years of my research. I am very thankful to Nick for sharing his immense knowledge and enthusiasm with me during our work. I appreciate our numerous talks and the time he spent patiently explaining me supergravity, string theory, black hole physics, differential geometry and many other topics. His excellent course on string theory helped me better understand the subject. I am grateful to Nick for suggesting interesting projects that enabled me to work on the spectrum of topics during my PhD. I also would like to thank Nick for arranging my multiple visits to IPhT-Saclay. Thanks to Nick for inviting me for dinners with his family and interesting conversations. Having Nick as my advisor has been a great experience during my time at USC. I am also very grateful to Krzysztof Pilch for collaboration on our papers. He has been a great source of knowledge for me on gauged supergravity and mathematics and was always willing to answer my questions. I equally enjoyed taking his class on methods of theoretical physics and assisting in teaching it later on. Thanks to Robert Walker for collaboration on our last paper that has become a part of this thesis. I also would like to thank Iosif Bena for arranging my visits at IPhT-Saclay and inviting me to Nick’s birthday conference. His many valuable suggestions and comments on my work were extremely helpful. I am also grateful to Iosif for his hospitality while I was staying there. I would like to extend my thanks to the members of the high energy physics group Clifford i Johnson, Krzysztof Pilch, Itzhak Bars, Dennis Nemeschansky, Nick Warner and Hubert Saleur for the great courses that enhanced my understanding of physics. Also thanks to Stephan Haas, DennisNemeschansky, KrzysztofPilch, NickWarner, DanielLidarandFeodorMalikovforkindly agreeing to participate in my qualifying exam and dissertation defense committees and providing their comments. I also want to thank my fellow graduate students Dmitry Rychkov, Orestis Vasilakis, Scott MacDonald, Ignacio Araya Quezada, Albin James, Vasilis Stylianou, Avik Chakraborty, Robert Walker and Felipe Rosso for the interesting talks and for making our group full of bright people. Finally, I wish to express my deepest gratitude to my parents for their continuous support and encouragement throughout all these years. ii Abstract In this thesis we investigate different aspects of supersymmetric microstate geometries and con- struct the renormalization group flows in holographic field theory. The idea of the microstate geometry program is to use the string theory as a consistent theory of quantum gravity to con- structively resolve the long-standing black hole information paradox and the related entropy problem by replacing the classical black hole with the horizonless geometry. The microstate geometries are smooth, horizonless solutions of classical supergravity equations which have the same conserved charges as black hole and allow for unitary scattering. There is a hope that one can find enough of them to account for the huge gravitational entropy of black hole at least at the semi-classical level. The construction of such microstates crucially relies on the presence of extra-dimensions, the non-trivial topology of spacetime and the long-ranged supergravity fluxes. In the first part we consider the supersymmetric renormalization group flows in gauged su- pergravity which share some of the properties of the black hole microstate geometries such as presence of the large topological cycles threaded by purely magnetic fluxes. The infra-red limit of the solutions appears to be singular from four-dimensional perspective; however, when uplifted to the full eleven-dimensional theory the solutions are remarkably regular. We demonstrate that the physical mechanism underlying this regularity is the same that allows one to resolve black hole singularities in microstate geometries and involves the polarization of M2 branes into M5 branes. We also generalize our flows to a new class of Janus solutions which describe the domain wall defects in the dual holographic field theory. In the second part we study the properties of the new class of excitations of microstates called iii W-branes that wrap the topological cycles inside the bubbled microstate geometries. First, we analyzethesupersymmetriesofthesolutionthatdescribestheindividualW-branesandshowthat in general it breaks all the supersymmetries of the corresponding black hole and therefore cannot be considered as a microstate. Second, we propose a new approach to supergravity description of W-brane condensates that can encode a large amount of black hole entropy. Our approach is based on extending the bubbled geometry to compactified extra-dimensions and putting them on the same dynamical footing with the spacetime. We find an exact solutions of BPS equations for backgrounds that have the same supersymmetries as the three-charged black hole. Finally, we examine the tidal forces acting on the radially infalling massive particle in the newly-constructed six-dimensional microstate geometry. These geometries have a long AdS throat that caps-off above the black hole would-be horizon. We find a remarkable difference between how the ordinary matter interacts with microstate compared to its interaction with the classical black hole. We predict that the probe particle must undergo highly energetic stringy transition before it reaches the cap of the geometry, and thus particle will ultimately scramble into microstate geometry. iv Contents Acknowledgments i Abstract iii 1 Introduction 1 2 AdS/CFT and holographic RG flows 10 2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Superconformal Chern-Simons-matter theories . . . . . . . . . . . . . . . . . . . . 12 2.2.1 N = 2 Chern-Simons-matter theory . . . . . . . . . . . . . . . . . . . . . . 12 2.2.2 N = 6 Chern-Simons-matter theory . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Gravitational description of coincident M2 branes . . . . . . . . . . . . . . . . . . 14 2.4 Holographic RG flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4.1 Flows in gauged supergravity . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4.2 Flows in ten-dimensional supergravity . . . . . . . . . . . . . . . . . . . . 19 2.5 RG flows in M-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3 Supersymmetric flows and Janus solutions in gauged supergravity and M- theory 23 3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 The truncation and BPS equations in four dimensions . . . . . . . . . . . . . . . . 24 3.2.1 The truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 v 3.2.2 Domain wall Ansätze and BPS equations . . . . . . . . . . . . . . . . . . . 27 3.2.3 Integrating the BPS equations . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2.4 Behavior at large λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 The uplift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3.1 SU(3)×U(1)×U(1) invariants on S 7 . . . . . . . . . . . . . . . . . . . . . . 35 3.3.2 The metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.3.3 Internal coordinates and local expressions . . . . . . . . . . . . . . . . . . 39 3.3.4 The transverse flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3.5 The space-time flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3.6 A summary of the uplift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4 The equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.4.2 The flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.4.3 The Einstein equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.4.4 The Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.5 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.5.1 Projector Ansätze . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.5.2 Supersymmetries for the Janus solutions . . . . . . . . . . . . . . . . . . . 60 3.5.3 The RG-flow limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.6 IR asymptotics in eleven dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.6.1 cos 3ζ6=−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.6.2 ζ =±π/3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.6.3 The IR limit of the flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4 Microstate geometries in five dimensions 76 4.1 Brane configuration and supersymmetries . . . . . . . . . . . . . . . . . . . . . . . 76 4.2 BPS equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 vi 4.3 Gibbons-Hawking space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.4 Magnetic fluxes on Gibbons-Hawking space . . . . . . . . . . . . . . . . . . . . . . 80 4.5 Solution of BPS equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.6 Closed time-like curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.7 Bubbled solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.8 Bubble equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5 Supersymmetry and wrapped branes in microstate geometries 89 5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.2 The Lagrangian and BPS equations . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.3 The standard bubbled geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.3.1 Two centers and AdS 3 ×S 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.3.2 Other frames and coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.3.3 Killing spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.4 The supersymmetries with hypermultiplet scalars . . . . . . . . . . . . . . . . . . 97 5.4.1 The hypermultiplet solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.4.2 The supersymmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6 Non-trivial compactifications with dynamical moduli 103 6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.2 Geometric preliminaries to reducing onM GH ×T 2 . . . . . . . . . . . . . . . . . 106 6.3 Generalities about Calabi-Yau compactification . . . . . . . . . . . . . . . . . . . 110 6.3.1 The form of the eleven-dimensional metric . . . . . . . . . . . . . . . . . . 110 6.3.2 Compensators and Lichnerowicz modes . . . . . . . . . . . . . . . . . . . . 111 6.3.3 Frames and spin connections . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.4 Reduction onM GH ×T 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.4.1 Lichnerowicz modes, connections and compensators onM GH . . . . . . . . 115 vii 6.4.2 The metric Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.4.3 The flux Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.4.4 The BPS equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.5 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.5.1 The BPS system, equations of motion and M2-brane sources . . . . . . . . 124 6.5.2 The Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.6 An example: Eguchi-Hanson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.6.1 Complex coordinates and trivializing the compensators . . . . . . . . . . . 131 6.6.2 The topological solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.6.3 Another effective action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7 Tidal stresses and energy gaps in microstate geometries 140 7.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 7.2 The microstate geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.2.1 The CFT states and dual geometries . . . . . . . . . . . . . . . . . . . . . 143 7.2.2 The family of metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 7.2.3 Limits of the metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 7.2.4 Some comments on curvatures and the supergravity approximation . . . . 147 7.3 The energy gap, red shifts and dispersion relations . . . . . . . . . . . . . . . . . . 150 7.3.1 Red shifts and E gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 7.3.2 Dispersion relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.4 Geodesics and probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.4.1 Radially infalling geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.4.2 Tidal forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 7.4.3 Redshifts and energy scales . . . . . . . . . . . . . . . . . . . . . . . . . . 162 7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 viii 8 Concluding remarks 165 Appendices 170 A Conventions 171 B Reduced E 7(7) tensors and the scalar action 173 C The Freund-Rubin flux 175 D The Ricci tensor 177 Bibliography 181 List of Figures 191 List of Tables 193 ix Chapter 1 Introduction Black holes are one of the most fascinating and mysterious objects in our Universe. They are regions of space with strong enough curvature that nothing cannot escape from inside, not even light. Classical black holes were first discovered in 1916 by Karl Schwarzschild [1] and were considered as purely theoretical possibilities for many decades. They are vacuum solutions of Einstein’s General Relativity and describe the gravitational field outside of any spherically symmetric object. The Schwarzschild solution is amazingly simple since it depends only on one parameter, the total mass of the body, M. Understanding the physics of black holes represents one of the greatest challenges in theoret- ical physics. The theory of black holes features the long-standing paradoxes, such as information paradox, and related black hole entropy problem. These problems are of great importance for contemporary physics since they lead to the inconsistency between the laws of classical general relativity and unitary quantum mechanics. The resolution of these puzzles is possible only in the framework of a more fundamental theory of gravity. String theory is a consistent theory of quantum gravity, and it should provide answers to these questions. Thus, black holes can be used as a very non-trivial test of the viability of string theory as a theory of quantum gravity. According to the Birkhoff’s theorem [2] the Schwarzschild black hole is the only static (time independent) vacuum solution with spherical symmetry. The black hole is characterized by the 1 event horizon, which is defined as a surface “of no return” past which the outgoing signal can never rich the future (null) infinity. The apparent singularity of metric at the event horizon turns out to be an artifact of the original Schwarzschild coordinates, and geometry can be extended beyond the horizon. The actual singularity, where the curvature of the spacetime becomes infinite, is located at the origin, r = 0. According to the classical theory the space under the horizon is vacuum, and all the stuff that made up a black hole ends up in the central singularity. The presence of a singularities obviously represents a problem for General Relativity, since it cannot describe physics at such points. The singularities are unavoidable in classical General Relativity. Using very general set of assumptions Hawking and Penrose proved that once the collapsing matter forms a trapped surface, it will evolve further to the singularity. The presence of singularities where General Relativity breaks down also shows the incompleteness of the theory and suggests that classical theory should be replaced by a more fundamental one which is able to resolve them. One may hope that for physically reasonable solutions the singularities are always hidden behind the event horizon and cannot be observed from future null infinity. The cosmic censor- ship conjecture asserts that naked singularity cannot form in gravitational collapse from generic, initially nonsingular states in asymptotically flat spacetime obeying the dominant energy condi- tion. In spite of many efforts the precise prove of cosmic censorship conjecture still remains an open problem of classical general relativity. Another remarkable property of four-dimensional black holes is their uniqueness [3]. The stationary asymptotically flat black hole coupled to the U(1) Maxwell field can be completely characterized by its conserved charges at infinity such as mass M, charge Q and angular mo- mentumJ. This theorem has been described more colloquially as “black holes cannot have hair” (“no-hair theorem”). In other words the moduli space of the classical black hole consists of just three parameters. This fact can be easily understood if one remembers that gravitational radia- tion in the leading order is caused by the change of quadrupole moment of the system. Therefore, if one considers the gravitational collapse of non-spherically symmetric object all higher multi- 2 pole charges will be radiated away in the form of gravitational waves, and the final state will be described by monopole and dipole charges only. Thus, the detailed information about the distribution of matter that formed the black hole is lost. In 1960’s and 70’s the famous theorems of black hole thermodynamics were established. Bekenstein argued [4] that black hole must have an entropy if we want to preserve the second law of thermodynamics. If one drops a box of hot gas into the black hole its entropy will be lost, and thus the total entropy of the Universe will decrease in contradiction with the second law. He argued that the black hole entropy must be proportional to its horizon area with universal coef- ficient and showed that the total change of entropy remains non-negative. Hawking later proved this conjecture and showed that black hole should evaporate in the presence of quantized matter fields [5,6]. He found that it emits radiation with spectrum of a black body at a temperature that is proportional to its surface gravity. That enabled Hawking to make Bekenstein’s identification between entropy and area precise and fix the constant of proportionality. However, that raised a question about microscopic origin of Bekenstein-Hawking entropy. If one computes the entropy of a solar mass black hole, one gets a huge number of 10 76 . The principles of statistical mechan- ics tells us that the entropy must be equal to the logarithm of the number of microstates. For the solar mass black hole that gives an enormous number of e 10 76 quantum microstates! These microstates cannot be found in the classical General Relativity because of the no-hair theorem, which state that black holes are unique and there is only one microstate, and therefore entropy should be zero. The classical thermodynamics tells us that entropy is an extensive quantity which is pro- portional to the volume of the system. The fact that the black hole entropy is proportional to the horizon area instead of its volume is highly surprising. It suggests that the fundamental degrees of freedom of black hole can be described by quantum field theory that lives in one dimension lower then the black hole. This idea enabled t’Hooft and Susskind [7,8] to formulate a holographic principle that was later realized in the form of the AdS/CFT correspondence [9]. Another related problem is the black hole information paradox. Hawking showed that the 3 vacuum near the horizon is unstable due to the creation of particle pairs: one of the virtual particles falls into the black hole and reduces its mass, while another one becomes on-shell and escapes to infinity. The black hole uniqueness makes Hawking radiation universal and almost featureless. Indeed, since Hawking radiation is created outside of the black hole its properties depend only on the conserved asymptotic charges of black hole. Once the black hole has completely evaporated the information about the initial state of matter that formed it will be lost. Moreover, this process violates one of the fundamental principles of quantum mechanics, the unitary evolution of states. If the original matter was in the pure state, it will eventually evolve into the final state of nearly thermal radiation, which is a mixed state. A statistical derivation of black hole entropy using string theory techniques was given first by Strominger and Vafa [10]. They reproduced the Bekenstein-Hawking formula for a special class of supersymmetric black holes in type IIB supergravity by counting the number of bound states of D-branes in the orbifold conformal field theory at vanishing string coupling. These states contribute to the supersymmetric index which is independent of the coupling. Thus, one can use the supersymmetry protection of index to compute the degeneracy of states at weak coupling, when the system is described by open strings ending on D-branes, and extend the result to the strong coupling, when the system is described by supergravity with backreacting D-branes forming a black hole. The other important step in understanding the microscopic structure of black holes was done with the aid of AdS/CFT correspondence. The class of black holes considered by Strominger and Vafa has a near-horizon limit of AdS 3 ×S 3 , so the duality is applicable. According to it the microstates of the black hole correspond to certain states in conformal field theory at the boundary ofAdS. Since the dynamics of the CFT is unitary the individual black hole microstates should preserve unitarity too, and thus the information paradox should be solved in principle. In 2009 using information theory Mathur proved a theorem [11] that resolution of the infor- mation paradox requires an “order 1” modification of physics at the horizon scale. In particular this implies that the information cannot be recovered by taking into account a higher order 4 quantum corrections to the semi-classical Hawking states, and one needs a macroscopic changes at the horizon. This idea was realized in the form of the fuzzball proposal for black holes which asserts that there are exponentially large number of horizonless configurations that asymptoti- cally look like the black hole but generically differ from the black hole up to the horizon scale. These solutions are considered as black hole microstates while the original black hole represents the average description of the system. Although it is believed that the most generic fuzzball will have regions of space with string-scale curvature and can only be described in the full string theory, one can try to use supergravity to find a generic enough solutions, that can account for the black hole entropy at the semi-classical level. The microstate geometry program extends the fuzzball proposal providing the only possible mechanism withing classical supergravity to support the necessaryO(1) changes at the horizon scale. A smooth horizonless solutions of supergravity which are valid within the supergravity approximation to string theory and have the same mass, charge and angular momentum as black holes are called microstate geometries. The construction of microstate geometries extensively relies on the non-trivial topology of spacetime and the presence of the long-range supergravity fluxes [12]. The mechanism by which microstate geometry resolves the singularity of black hole is known as a geometric transition, which means that the classical singularity is blown up and replaced by non-trivial cycles threaded by cohomological fluxes. The resulting bubbled geometry becomes completely regular with no horizon and has the same charges as black hole. The geometric transition has been successfully used to resolve naked singularities that appear in other areas of holographic field theories. The most famous example is the Klebanov-Strassler solution [13] which is a gravity dual of renormalization group flow inN = 1 supersymmetric gauge theory. The earlier solution of Klebanov and Tseytlin [14] contained a singular conifold which could not correctly describe the infrared physics. Klebanov and Strassler showed how the correct infrared phase involves chiral symmetry breaking and confinement. In supergravity chiral symmetry breaking is realized through blowing up the cycle of finite size at the bottom of the conifold. The resulting geometry, a deformed conifold, is completely nonsingular and without a 5 horizon. Other examples of bubbled geometries include Lin-Lunin-Maldacena [15], Polchinsky- Strassler [16] and Bena-Warner [17] solutions. Thus, studying smooth supergravity solutions supported by fluxes plays an important role both for discovery of new infrared phases of matter and for building new microstates of black holes. The AdS/CFT correspondence provides a valuable tool for studying the strongly coupled quantum systems. Combined with supersymmetry it provides analytic control over holographic renormalization group flows and enables the analysis of non-perturbative, infrared structure of the theory. The first part of this thesis is devoted to study the holographic description of renormalization group flows. In Chapter 2 we provide a relevant background by giving a brief review of the superconformal Chern-Simons-matter theory known as ABJM theory, which describes the low- energy dynamics of M2 branes, and its gravitational description in terms of M-theory onAdS 7 × S 4 /Z k . We also discuss the renormalization group flow driven by deformations of the conformal theory by relevant operator, which are dual through the AdS/CFT correspondence to the domain wall solutions in maximal gauged supergravity. In Chapter 3 we investigate a family of SU(3)×U(1)×U(1)-invariant holographic flows and Janus solutions obtained from gaugedN =8 supergravity in four dimensions. We give complete details of how to use the uplift formulae to obtain the corresponding solutions in M-theory. While the flow solutions appear to be singular from the four-dimensional perspective, we find that the eleven-dimensional solutions are much better behaved and give rise to interesting new classes of compactification geometries that are smooth, up to orbifolds, in the infra-red limit. Our solutions involve new phases in which M2 branes polarize partially or even completely into M5 branes. The presence of magnetic fluxes of M5 branes prevents singularities and makes the infrared limit of the flow solution smooth. This feature closely resembles that of black hole microstategeometries. Wealsoderivetheeleven-dimensionalsupersymmetriesandshowthatthe eleven-dimensional equations of motion and BPS equations are indeed satisfied as a consequence of their four-dimensional counterparts. 6 In the second part we consider different aspects of the microstate geometries of black holes. In Chapter 4 we review the construction of supersymmetric microstates of five-dimensional black hole with three charges. Starting with M-theory on six-torus we consider the brane configuration that has the same supersymmetries as the black hole, formulate corresponding BPS equations in five dimensions and describe the procedure of finding a regular solution. We end the review with a discussion of the issue of closed time-like curves (CTC’s) and derive the necessary conditions for their absence known as bubble equations. It has been recently proposed that a significant amount of black hole entropy can be en- coded in the condensates of W-branes wrapping the topologically non-trivial cycles in microstate geometries [18]. It has also been suggested that these warped branes themselves might be consid- ered as BPS black hole microstates. In Chapter 5 we study the supersymmetries of the solution proposed in the literature that describes M2 branes warping a cycle inside 1 8 -BPS microstate ge- ometry. We show that such brane wrappings will generically break all the supersymmetries. In particular, all the supersymmetries will be broken if the branes wrap cycles that are not co-linear or if the net charge of the wrapped branes is zero. We show that if M2 branes wrap a single cycle, or if they wrap a several of co-linear cycles with the same orientation, then the solution will be 1 16 -BPS, having two supersymmetries. We conclude that the proposed solution cannot be the black hole microstate since it breaks some or all of its supersymmetries. The standard microstate geometries were obtained using theT 6 compactification of M-theory to five-dimensional supergravity coupled to two vector multiplets. This analysis was sufficient to find a broad class of microstatate structures using exact solutions of supergravity equations. The topology of compactification manifold T 6 remained fixed and played an auxiliary role in this construction. The physics of W-branes adds a new level of complexity into this picture by introducing a non-trivial sources and dynamics on the compactified directions: from the viewpoint of compact space W-branes look like a charged particles moving in a magnetic field. To describe such microstate structure with fully back-reacted details one has to go beyond the toroidal compactification and use more general Ricci-flat manifolds with non-trivial topology. 7 Speaking more generically one would ultimately need to put the space-time manifold and the compactification manifold on the same dynamical footing. In Chapter 6 we suggest a possible approach to finding the correct supergravity description of wrapped brane condensates based on adding a new dynamical structures in compactified directions. We consider exact, eleven-dimensional, BPS supergravity solutions in which the compactificationinvolvesanon-trivialCalabi-YaumanifoldwithdynamicalKählermoduli. Since there are no explicitly-known metrics on non-trivial, compact Calabi-Yau manifolds, we use a non-compact “local model” and take the compactification manifold to beY = M GH × T 2 , whereM GH is a hyper-Kähler, Gibbons-Hawking ALE space. We focus on eleven-dimensional backgrounds that compactify to five-dimensional black holes with three electric charges. We find exact families of solutions to the BPS equations for our non-compact “local model” that have the same four supersymmetries as the three-charge black hole. We demonstrate that in order to solve BPS system, the Calabi-Yau manifold has to be fibered over the spatial base using compensating fields onY. Their role is to ensure smoothness of the eleven-dimensional metric when the moduli ofY depend on the space-time. We also examine the equations of motion for flatR 4 base and discuss the brane distributions on generic internal manifolds that do not have enough symmetry to allow smearing. As we have mentioned before microstate geometries provide a necessaryO(1) corrections to the black hole physics at the horizon scale. General Relativity predicts that the tidal stress on an infalling observer can be made arbitrary small by increasing the mass of the black hole. Thus in classical theory the observer will not experience any dramatic impact until one reaches the central singularity. It is therefore extremely interesting to see how these new large-scale structures affect the ordinary matter, and whether one can find some new physical effects near the horizon associated with microstates. In Chapter 7 we study the motion of radially infalling massive probe particle in a family of newly-constructed microstate solutions in six dimensions. These geometries have long AdS throats that cap-off arbitrary close to where the black hole horizon would be. They also have a 8 precisely known holographic description in terms of the states of dual D1-D5 CFT. In contrast with theclassical predictions wefind thatin thedeepestgeometries, whichhavethe lowest energy gaps, the geodesic deviation shows that the stress reaches the Planck scale long before the probe reaches the cap of the geometry. Such probes must therefore undergo a stringy transition as they fall into microstate geometry. We discuss the scales associated with this transition and comment on the implications for scrambling in microstate geometries. Finally in Chapter 8 we give our concluding remarks and outline the directions for further investigations. This thesis is based on the papers I wrote in collaboration with Krzysztof Pilch, Nicholas Warner and Robert Walker [19–23]. 9 Chapter 2 AdS/CFT and holographic RG flows 2.1 Motivation The AdS/CFT correspondence [9,24,25] provides a valuable tool for studying the strongly cou- pled quantum systems. It relates the string/M-theory in (d + 1)-dimensional Anti-de-Sitter (AdS) space to the conformal field theory (CFT) without gravity in d dimensions. The most elaborate example of AdS/CTF correspondence relates the theory of coincident D3 branes to four-dimensionalN = 4 supersymmetric Yang-Mills theory in four dimensions. The low en- ergy limit of string theory is described by classical supergravity, and the stack of N D3 branes generates the AdS 5 ×S 5 geometry in the near-horizon limit. Thus, type IIB supergravity on AdS 5 ×S 5 corresponds toN = 4 super-Yang-Mills. Both theories provide a dual description of the same physics, so that when the theory on one side of the duality becomes strongly coupled, the other one is in weak coupling regime. This enables to study the non-perturbative effects in gauge theory using weakly coupled supergravity. Another example of the AdS/CFT correspondence comes from coincident M2 banes in M- theory. Their low energy dynamics is described by (2 + 1)-dimensional superconformal Chern- Simons theories coupled to massless matter. The theories with extendedN = 6 supersymmetries were developed in a series of papers by [26–30]. The particular Chern-Simons theory with 10 U(N)×U(N) gauge group and levels (k,−k) is known as ABJM theory [30]. This theory has received wide attention, for instance, in the context of the AdS/CMT correspondence which allows to use M-theory to describe strongly coupled condensed matter systems. AdS/CFT correspondence can be used to study not only the conformal field theories but also their deformations. If one deforms the theory in the UV by a relevant operator the conformal symmetry will be broken, and the theory will start flowing into the IR. The holographic renor- malization group flows are the gravitational duals of the regular RG flows in the field theory. Studying the properties of gravitational flow solutions allows us to analyze the infrared phases of the dual holographic matter. In gauged supergravity holographic RG flows are described by the domain wall solutions that interpolate between different critical points of the scalar potential. Such flows involve turning on particular combinations of bosonic and fermionic bilinear operators in field theory which make it flow to the conformal fixed point in the IR. These flows are totally smooth in the dimensionally reduced theory, and their infrared geometry is a regular AdS×S spacetime. However, the flows between critical points are quite rare, and most of the flow develop a singularity in the infrared end of the flow. There is wide class of singular flows known as Coulomb branches which has a clear physical interpretation. Coulomb branch flows correspond toSU(N) gauge symmetry being broken, typically, toU(1) N , and their infrared limit is described by branes spreading out into some distribution in space. For example, the gravity duals of the Coulomb branches inN = 4 super-Yang-Mills have a naked singularity from the five-dimensional perspective, but they usually become less singular when uplifted to ten dimensions [31]. It was shownthatsuchstatescorrespondtothegeometrywhichisthenear-horizonlimitofacontinuous distribution of D3 branes. Despite this Coulomb branch flows still have a singularity at the points where the electrically charged sourced are located. Therefore it is extremely interesting to see if one can find other types flows which do not go to the critical point but have a regular geometry in the IR limit. In Chapter 3 we describe a family of holographic RG flow in M-theory with the novel inter- 11 esting features. The infrared limit of our flows look singular in the four-dimensional theory, but the uplift to eleven dimensions appears to be remarkably regular. The flows go from (2 + 1)- dimensional theory on M2 branes in the UV to a (3 + 1)-dimensional theory on intersecting M5 branes in the IR. Our solutions also exhibit the properties similar to that of black-hole microstate geometries. In particular, the IR geometry of our flows is sourced by the magnetic fluxes of M5 branes through Chern-Simons interactions. This is very reminiscent of the mechanism called “charges dissolved in fluxes” which is a key ingredient of the microstate geometry program and enables to resolve the black hole singularity to obtain a smooth, horizonless solution. The purpose of this chapter is to introduce the ABJM field theory and its dual formulation in terms of M-theory on AdS 4 ×S 7 /Z k whose deformations are described by our holographic flows in Chapter 3. We follow the original paper of ABJM and notes [32] where other references to the literature can be found. Then we also illustrate the method of holographic flows by describing the known example of smooth RG flow solution in five-dimensional gauged supergravity and provide a formula for the uplift of its metric to type IIB supergravity. The chapter ends with the discussion of RG flows in M-theory. 2.2 Superconformal Chern-Simons-matter theories 2.2.1 N = 2 Chern-Simons-matter theory Chern-Simons theories in 2+1 dimensions are purely topological theories. When they are coupled to matter fields, they are no longer topological, but they still can retain the conformal symmetry. N = 2 theories in three dimensions can be obtained fromN = 1 theories in four dimensions by dimensional reduction if one replaces the kinetic term of the vector field by supersymmetric Chern-Simons term. The field content of such theories includes a vector multiplet V in the adjoint of the gauge group G, composed of gauged field A μ , two-component Dirac spinor χ, a scalar field σ, which comes from the A 3 component of gauge field when dimensionally reduced from 3 + 1 dimensions, and auxiliary scalar field D. The Chern-Simons action written in the 12 component fields in Wess-Zumino gauge has the form S N =2 CS = k 4π Z Tr(A∧dA + 2 3 A 3 − ¯ χχ + 2Dσ). (2.1) where constant k is called the Chern-Simons level. The invariance under large gauge transfor- mations of non-abelian theory restricts k to integers. The theory can be coupled toN f chiral multiplets Φ i in representationR ofG which consists ofscalarφ i andfermionψ i . Thekinetictermofthechiralmultipletisobtainedbythedimensional reduction from four dimensional kinetic term. The matter action is S matter = Z N f X i=1 (D μ ¯ φ i D μ φ i +i ¯ ψ i γ μ D μ ψ i − ¯ φ i σ 2 φ i + ¯ φ i Dφ i − ¯ ψ i σψ i +i ¯ φ i ¯ χφ i −i ¯ ψ i χψ i ). (2.2) The action has U(N f ) global flavor symmetry. The auxiliary fields σ and D can be integrated out using their equations of motions. The resulting classical action is marginal. It has been argued that the action does not receive quantum corrections beyond the one-loop shift of k, and conformal symmetry of the action is preserved at quantum level. 2.2.2 N = 6 Chern-Simons-matter theory TheN = 6theoryisaspecialversionoftheN = 2theory. ItwasobtainedbyAharony, Bergman, Jafferis and Maldacena [30] and is known as ABJM theory. The theory is based on the gauge group U(N)×U(N). It has two chiral superfields, A 1 ,A 2 , in the bifundamental representation and two anti-bifundamental chiral superfields,B 1 ,B 2 . The Chern-Simons levels of the two gauge groups are chosen to be equal but opposite in sign. Integrating out the auxiliary fieldsφ i , we get superpotential W = 4π k Tr(A 1 B 1 A 2 B 2 −A 1 B 2 A 2 B 1 ). (2.3) 13 The superpotential can also be written as W = 2π k ab ˙ a ˙ b Tr(A a B ˙ a A b B ˙ b ), (2.4) which exhibits large SU(2)×SU(2) symmetry acting separately on A’s and B’s. All other terms in the action have this bigger symmetry, so the full Chern-Simons-matter theory has this enhanced symmetry. The important property of classical ABJM is that U(1) R symmetry combines with flavor SU(2)×SU(2) into a large SU(4) R symmetry, which corresponds at least toN = 6 supersym- metry. The particular version of ABJM theory with SU(2)×SU(2) gauge group and levels k = 1, 2 was constructed earlier by Bagger and Lambert [26–28], and Gustavsson [29] and is known as BJL.TheBJLmodelfeaturesN = 8supersymmetry, andthereforehasSO(8) R symmetrygroup. The coupling constant of the theory is 1/k, so that it is weakly coupled at largek. In the large N limit the effective coupling constant in the planar diagrams is the ’t Hooft coupling λ≡N/k. Thus, the theory becomes weakly coupled for k N and strongly coupled for k N. The ’t Hooft limit of largeN has a dual description in terms of weakly coupled string theory as we will discuss below. 2.3 Gravitational description of coincident M2 branes ABJM theory is directly related to M-theory. The low energy limit of M-theory is known to be N = 1 supergravity in eleven dimensions which is a unique supergravity theory. It consists of one graviton, g μν , a gravitatino,Ψ μ , and the tree-index skew-symmetric gauge field A μνρ whose field strength is the four-form F 4 . 14 A stack on N coincident M2 branes in flat space creates the following extremal geometry: ds 2 11 = h(r) −2/3 (−dt 2 +dx 2 1 +dx 2 2 ) +h(r) 1/3 (dr 2 +r 2 dΩ 2 7 ), h(r) = 1 + L 6 r 6 , (2.5) F 4 = dt∧dx 1 ∧dx 2 ∧dh(r) −1 , where N is the number of M2 branes and is related to L by L 6 = 32π 2 Nl 6 p . In the limit r→ 0, the metric becomes that of AdS 4 ×S 7 : ds 2 11 = L 2 1 4 ds 2 AdS 4 +ds 2 S 7 . (2.6) If we consider a stack of M2 branes placed at the singularity of the conifoldC/Z k , which is described by the levelk ABJM theory, thendΩ 2 7 refers to the metric on unitS 7 /Z k . One can use four complex coordinates z i (i = 1, 2, 3, 4) to describe the transverse to M2 brane space. Then Z k identification acts as z i →e i 2π k z i . (2.7) SinceZ k preserves anSU(4)×U(1) isometry acting onz i in an obvious way, it is natural to use the description of S 7 as an S 1 fibration overCP 3 . The Hopf fibration of S 7 is ds 2 7 = (dϕ 0 +ω) 2 +ds 2 CP 3, ds 2 CP 3 = P i dz i d¯ z i ρ 2 − | P i z i d¯ z i | 2 ρ 4 , ρ 2 ≡ 4 X i |z i | 2 , dϕ 0 +ω ≡ i 2ρ 2 (z i d¯ z i − ¯ z i dz i ). (2.8) dω = J = id z i ρ ! ∧d ¯ z i ρ ! , where ϕ 0 is periodic with period 2π and J is proportional to the Kähler form ofCP 3 . TheZ k 15 quotient is done by writing ϕ 0 =ϕ/k, with ϕ∼ϕ + 2π. The quotient metric becomes ds 2 S 7 /Z k = 1 k 2 (dϕ +ω) 2 +ds 2 CP 3. (2.9) To have a properly quantized flux,F 4 has to be proportional toN 0 =kN instead ofN, since the volume of the space become reduced by factor 1/k. The radius of theCP 3 is large whenN 0 1. The radius of the S 1 circle in Plank units is of order L/kl p ∝ (Nk) 1/6 /k. Thus, the M-theory description is valid if k 5 N. 2.4 Holographic RG flows The deformation of superconformal field theories by relevant operator breaks some of its sym- metries and make the theory flow with renormalization group (RG) from ultraviolet (UV) fixed point to infrared (IR). An example of the holographic RG flow which has a clear field-theoretical interpretation was constructed by Freedman, Gubser, Pilch and Warner (FGPW flow) [33]. They considered a deformation ofN = 4 super-Yang-Mills by giving the mass to one of three chiral superfields. The mass term breaks conformal invariance and drives the RG flow. In the field theory this type of deformation is known as Leigh-Strassler deformation [34]. The RG flow pre- servesN = 1 supersymmetry and restores the conformal invariance in the infrared fixed point. The authors of [33] argued that the holographic dual of the Leigh-Strassler flow is described by the domain wall solution that interpolates between the AdS geometries at two critical points of five-dimensional maximalN = 8 gauged supergravity. The evidence for the correspondence between the field theory flow and the supergravity domain wall was established by matching the symmetries along the flow and by matching of the trace anomaly coefficients at both endpoints in field theory and through the AdS/CFT correspondence. The five-dimensional maximalN = 8 gauged supergravity is a consistent truncation of the ten-dimensional type IIB supergravity. The consistent truncation means that any solution of the truncated theory can be uplifted to a solu- tion of the untruncated theory. The uplift of FGPW flow to IIB supergravity was later obtained 16 in [35]. 2.4.1 Flows in gauged supergravity The Lagrangian of gaugedN =8 supergravity has 42 scalar fields φ I that parametrize the coset E 6(6) /USp(8) with a complicated potential V (φ I ). The critical points of V correspond to AdS 5 solutions of gauged supergravity. The potential does not depend on the dilaton and axion fields which makes it a function of 40 variables. In practice it is impossible to extremize Vover such large number of parameters, so one has to use an invariant truncation to some subsector of the original theory. The invariant truncation consists of fixing the particular subgroup of SO(6) R-symmetry group and taking only those scalar fields, which are singlets with respect to this subgroup. It follows from group theory that the expansion of the potential around its critical point can only be quadratic in non-singlet fields. Thus, the stationary point of the potential on restricted subset of scalars will be the stationary point of the original potential. The critical points ofN = 8 gauged supergravity potential preserving a particular SU(2) symmetry were classified in [36]. It was shown that due to the residual invariance the scalar potential, V, can be parametrized by four real variables, ϕ 1 , ϕ 2 , φ and α: V = − g 2 4 ρ −4 1− cos 2 (2φ) (sinh 2 (ϕ 1 )− sinh 2 (ϕ 2 )) 2 + ρ 2 (cosh(2ϕ 1 ) + cosh(2ϕ 2 )) + 1 16 ρ 8 2 + 2 sin 2 (2φ) − 2 sin 2 (2φ) cosh(2(ϕ 1 −ϕ 2 ))− cosh(4ϕ 1 )− cosh(4ϕ 2 ) , (2.10) where ρ =e α . On the subset of scalars ϕ 3 = √ 6α the potentialV is related to the quantity called superpo- tential W by V = g 2 8 3 X j=1 ∂W ∂ϕ i 2 − g 2 3 |W| 2 , (2.11) 17 where W =λ 1 or W =λ 2 , and λ 1,2 are the eigenvalues of the tensor W ab : λ 1 = − e −2iφ 4ρ 2 h ρ 6 (2 cos(2φ)− cosh(2ϕ 1 ) + cosh(2ϕ 2 ) + 2i sin(2φ) cosh(ϕ 1 −ϕ 2 )) + (2 cos(2φ)(cosh(2ϕ 1 ) + cosh(2ϕ 2 )) + 4i sin(2φ) cosh(ϕ 1 +ϕ 2 )) i , λ 2 = − e −2iφ 4ρ 2 h ρ 6 (2 cos(2φ) + cosh(2ϕ 1 )− cosh(2ϕ 2 ) + 2i sin(2φ) cosh(ϕ 1 −ϕ 2 )) + (2 cos(2φ)(cosh(2ϕ 1 ) + cosh(2ϕ 2 )) + 4i sin(2φ) cosh(ϕ 1 +ϕ 2 )) i . (2.12) To find the supersymmetric bosonic background one has to set the supersymmetry variations of fermionic fields to zero. TheN =2 supersymmetric ground states were found to be: ϕ 1 = ± 1 2 log(3) , ϕ 2 = 0, ϕ 3 ≡ √ 6 α = 1 √ 6 log(2) , φ = 0. (2.13) The solution will flow the central critical point with maximal supersymmetry to the above critical point. The domain wall ansatz for the bulk metric that preserves Poincaré symmetry in four dimen- sions has the form: ds 2 1,4 =e 2A(r) η μν dx μ dx ν +dr 2 . (2.14) If A(r) = r l the metric becomes the one of anti-de Sitter space. The holographic RG flow interpolates between the ultravioletAdS 5 region atr→∞ and infraredAdS 5 region atr→−∞. To preserveN = 1 supersymmetry on the branes all along the flow, the domain wall has to be supersymmetric itself. The supersymmetry conditions impose φ = 0 leading to the system of equations dϕ j dr = g 2 ∂W ∂ϕ j , (2.15) A 0 = − g 3 W. (2.16) It is consistent with supersymmetry to set ϕ 2 = 0 along the flow, which leads to the extra 18 -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 2.1: The contour map of V (on the left) and W (on the right), with ϕ 1 on the vertical axis and α = 1 √ 6 ϕ 3 on the horizontal axis. The five labeled points are the only extrema of V in this plane. A numerical solution of the steepest descent equations is shown superimposed on the contour plot of W [33]. U(1) symmetry at the infrared fixed point. The superpotential reduces to W = 1 4ρ 2 cosh(2ϕ 1 )(ρ 6 − 2)− (3ρ 6 + 2) . (2.17) The contour maps of V and W on the (α = 1 √ 6 ϕ 3 ,ϕ 1 ) parameter space are shown in Figure 2.1. The map of V shows five extrema. TheN =2 supersymmetric points 4 and 5 are SU(2)×U(1) invariant points related byZ 2 symmetry. The supersymmetric domain wall flows from central maximum ofW at (0, 0) to saddle point 4, which is dual to superconformal Leigh-Strassler point. The flow was obtained by solving numerically the system (2.16) with proper initial conditions in the UV point. This flow is going from AdS 5 in the UV to another AdS 5 in the IR and is completely smooth solution in five dimensions. 2.4.2 Flows in ten-dimensional supergravity Most of the flow solutions starting at the central point of maximal symmetry run to infinite values of the supergravity potential. Such flows were called “Flows to Hades”. From the point of view of five-dimensional supergravity these flows are pathological since the metric develops a singularity. Typically one finds that scalars ϕ j and function A(r) diverge logarithmically which 19 results in the vanishing cosmological term at some finite r. However, when uplifted to ten- dimensions, the solutions of IIB supergravity usually become less singular and may have simple physical interpretation. This softening of the five dimensional singularity happens because the ten-dimensional metric involves multiplying the 5-metric by a warp-factor, which modifies its asymptotic behaviour. The simple example is the Coulomb-branch flows considered in [31,37, 38]. In five dimensions the metric had a singularity at r = 0, whereas the corresponding ten- dimensional metric becomes singular at the location of sources which correspond to a continuous distribution of D3-branes. Thus, to understand the infrared limit of such a holographic RG flow properly one needs to uplift the solution to ten-dimensional theory. The uplift of the FGPW flow [33] was obtained in [35]. It was argued in [36] that the inverse metric, g pq , on the deformed S 5 is given by Δ − 2 3 g pq = 1 a 2 K IJp K KLp ˜ V IJab ˜ V KLcd Ω ac Ω bd , (2.18) whereV = (V IJab ,V Iα ab ) is the scalar matrix of the coset E 6(6) /USp(8) and ˜ V = ( ˜ V IJab , ˜ V Iα ab ) is the inverse ofV, K IJp are Killing vectors on S 5 , Ω ab is the USp(8) symplectic form, and Δ = det 1/2 (g mp ◦ g pq ), where ◦ g pq is inverse of the round S 5 metric. The quantity Δ can be obtained by taking the determinant of both sides of (2.18). The ten-dimensional metric is then given by ds 2 10 = Ω 2 ds 2 1,4 +ds 2 5 , (2.19) where ds 2 1,4 in the metric ofN = 8 supergravity in five dimensions, ds 2 5 = g mn dy m dy n is the deformed S 5 metric given by (2.18), and Ω 2 = Δ − 2 3 is the warp factor. The internal metric for the Leigh-Strassler flow which was discussed in subsection 2.4.1 and involves two scalars α and ϕ 1 was found in [35]. In the Cartesian coordinates on R 6 with S 5 20 given as a unit sphere, P (x I ) 2 = 1, the internal metric is: ds 2 5 (α,ϕ 1 ) = a 2 2 sechϕ 1 ζ (dx I Q IJ dx J ) + a 2 2 sinhϕ 1 tanhϕ 1 ζ 3 (x I J IJ dx J ) 2 , (2.20) where Q is a diagonal matrix with Q 11 = ... = Q 44 = e −2α and Q 55 = Q 66 = e 4α , J is an antisymmetric matrix with J 14 =J 23 =J 65 = 1, and ζ 2 =x I Q IJ x J . The warp factor is Ω 2 = ζ coshϕ 1 . (2.21) The complete solution including fluxes can be found using field equations and underlying sym- metry, which was done in [35]. 2.5 RG flows in M-theory If one starts with the field theory on a large stack of M2 branes then the holographic dual is simplyAdS 4 ×S 7 , and turning on fermion masses or vevs of fermion bilinears necessarily means turning on magnetic M5 brane fluxes on S 7 . This leads to some very interesting holographic renormalization group flows. For instance, one can turn on a particular set of fermion and bo- son mass terms and flow to conformal fixed points. Such flows have been extensively studied using four-dimensional,N =8 gauged supergravity [39–45] and involve turning on scalar fields in supergravity and moving between different critical points of the scalar potential in four dimen- sions [46,47]. Uplifts to eleven dimensions for some of these solutions have been obtained and analyzed in [48,49]. The Coulomb-branch flows for which, in the infra-red limit, the M2 branes spread out into some distribution in space were considered in [50–52]. GaugedN = 8 supergravity in four dimensions was first constructed in [53] and there was a great deal of subsequent work that argued how this must be related to the S 7 -compactification of M-theory. Over the years it has become evident that the gauged theory is indeed a consistent truncation and formulae have emerged showing precisely how gauged supergravity encodes so- 21 lutions to M-theory. One of the first general formula was given in [54] where it was shown how to compute the exact deformed metric on the S 7 in terms of all the supergravity scalars. This knowledge alone was immensely useful in finding uplifted solutions explicitly [41,42,48]. Exact formulae for fluxes proved to be a much greater challenge. It is only recently that a new set of considerably more workable uplift Ansätze for the internal 3-form potential have been proposed in [55,56] and then extended to the other components of the flux [57–60]. In the next chapter we investigate a family ofSU(3)×U(1)×U(1)-invariant holographic flows and related Janus solutions obtained from gauged N=8 supergravity in four dimensions. We give complete details of how to use the uplift formulae to obtain the corresponding solutions in M- theory. While the flow solutions appear to be singular from the four-dimensional perspective, we find that the eleven-dimensional solutions are much better behaved and give rise to interesting new classes of compactification geometries that are smooth, up to orbifolds, in the infra-red limit. Our solutions involve new phases in which M2 branes polarize partially or even completely into M5 branes. We derive the eleven-dimensional supersymmetries and show that the eleven- dimensional equations of motion and BPS equations are indeed satisfied as a consequence of their four-dimensional counterparts. Apart from elucidating a whole new class of eleven-dimensional Janus and flow solutions, our work provides extensive and highly non-trivial tests of the uplift formulae. 22 Chapter 3 Supersymmetric flows and Janus solutions in gauged supergravity and M-theory 3.1 Motivation In this chapter we describe the holographic RG flow solutions that start on M2 branes in the UV and go to solutions sourced largely, or even entirely by M5 branes in the IR. Unlike many flows to the IR, these flows, when uplifted, have only mild orbifold singularities. Moreover, there is a special class of flows that go to pure M5 branes and can be interpreted as describing an “almost conformal” fixed point in (3 + 1) dimensions. We will also look at Janus solutions that delve into the backgrounds described by the flows and so may be interpreted as describing interfaces between phases described by the holographic IR flows. The chapter is organized as follows. In section 3.2 we consider the SU(3)×U(1)×U(1)- invariant truncation of four-dimensionalN = 8 gauged supergravity, which involves only one scalar and one pseudoscalar fields, and compute their scalar potential and superpotential. We obtain the first order equations equations that describe the supersymmetric flows and integrate 23 them in full generality. We also analyze the infrared asymptotic of our flow solution. In section 3.3 we describe the details of how this sector of gauged supergravity is uplifted to M-theory. We give the uplift of the metric and flux to eleven dimensions. Subsection 3.3.6 contains the summary of the uplifted fields. In section 3.4 we show how this uplifted solution solves the equations of motion of M-theory. The supersymmetry structure of the solutions is analyzed from the eleven-dimensional perspective in section 3.5. In section 3.6 we study the IR limits of our holographic flows and how they are related to distributions of M5 and M2 branes. We discuss the limiting metric of the particular RG flow and its possible interpretation in terms of the dual field theory. 3.2 ThetruncationandBPSequationsinfourdimensions 3.2.1 The truncation In this section we summarize some explicit results for the truncation of four-dimensional,N =8 supergravity [53] to the SU(3)×U(1)×U(1)-invariant sector that we will need for the uplift to M theory in section 3.3. Our discussion here is based on [61] and [19]. The Lagrangian for the truncationcanalsoberead-offfromamoregeneral SU(3)-invarianttruncationin[47]and[44,45]. The SU(3)×U(1)×U(1)⊂ SO(8) symmetry group of the truncation is defined by its action on the supersymmetries, i , of theN = 8 theory. We choose SU(3) and the first U(1) to act on the indices i = 1,..., 6, while the second U(1) on the indices i = 7, 8. This corresponds to the branching 8 v −→ (3, 1, 0) + (3,−1, 0) + (1, 0, 1) + (1, 0,−1). (3.1) The resulting truncation is particularly simple since, as observed in [61], the commutant of the symmetry group in E 7(7) consists of a single SL(2,R). The invariant fields are: the graviton,g μν , the gauge field, A α μ , for the two U(1)’s, a scalar, x, and a pseudoscalar, y. As we will describe below, this may be viewed as the bosonic sector ofN = 2 supergravity coupled to a vector 24 multiplet. The two non-compact generators of SL(2,R) in the fundamental representation of E 7(7) can be chosen as follows: T s =     0 Φ + IJKL Φ + IJKL 0     , T c =     0 i Φ − IJKL −i Φ − IJKL 0     , (3.2) where Φ ± IJKL = 24 (δ 1234 IJKL +δ 1256 IJKL ±δ 1278 IJKL +δ 3456 IJKL ±δ 3478 IJKL ±δ 5678 IJKL ), (3.3) are self-dual (+) and antiself-dual (−) SO(8) tensors, respectively. Then the scalar ‘56-bein’ is V ≡ e V =     u ij IJ v ijIJ v ijIJ u ij IJ     , V = xT s +yT c , (3.4) where the scalar, x≡λ cosζ, and the pseudoscalar, y≡λ sinζ, parametrize the coset SL(2,R) SO(2) , (3.5) with the canonical complex coordinate, z, given by z = tanhλe iζ . (3.6) Given the explicit generators (3.2), it is easy to check that the exponential (3.4) reduces to a polynomial, V = a 0 +a 1 V +a 2 V 2 +a 3 V 3 , (3.7) where a 0 = 2− 3|z| 2 2(1−|z| 2 ) 3/2 , a 1 = 6− 7|z| 2 6(1−|z| 2 ) 3/2 , a 2 = 3a 3 = 1 2(1−|z| 2 ) 3/2 , (3.8) 25 Note that the order of this polynomial coincides with the index of embedding of SL(2,R) in E 7(7) . Using the 56-bein (3.4), it is now straightforward to obtain the full bosonic action of the truncated theory [45]. In particular, we find that it is consistent to set the vector fields, A α μ , to zero. Then the resulting Lagrangian for the gravity coupled to the scalar fields is: 1 e −1 L = 1 2 R− 3 ∂ μ z∂ μ ¯ z (1−|z| 2 ) 2 − 6g 2 1 +|z| 2 1−|z| 2 = 1 2 R− 3∂ μ λ∂ μ λ− 3 4 sinh 2 (2λ)∂ μ ζ∂ μ ζ + 6g 2 cosh(2λ). (3.9) The Lagrangian (3.9) has no explicit dependence on the phase, ζ, and hence there is a conserved Noether current J μ = e sinh 2 (2λ)∂ μ ζ, (3.10) with the corresponding U(1) ζ symmetry being simply a rotation between the scalar and the pseudoscalar. It was shown in [45,61] that by keeping the SU(3)-invariant fermions, the truncation yields a N =2 supergravity in four dimensions. Its R-symmetry is a combination of the two U(1)’s and, from the supersymmetry variations, δψ i μ = 2D μ i + √ 2gA a ij γ μ j , i,j = 7, 8, (3.11) the real superpotential, W, is given by an eigenvalue of the A 1 -tensor, W = √ 2|A 1 77 | = √ 2|A 1 88 |, see appendix B. Substituting the real fields, λ and ζ, in (B.4), we then find W = √ 2 q sinh 6 λ + cosh 6 λ + 2 sinh 3 λ cosh 3 λ cos(3ζ). (3.12) In terms of the superpotential, W, the potential P = − 6 cosh(2λ), (3.13) 1 See, [44,45] and appendix B. 26 is given by P = 1 3 " ∂W ∂λ ! 2 + 4 sinh 2 (2λ) ∂W ∂ζ ! 2 − 3W 2 . (3.14) Note that unlike the potential,P, the superpotential, W, is invariant only under aZ 3 subgroup of U(1) ζ . 3.2.2 Domain wall Ansätze and BPS equations In this chapter we are interested in a special class of solutions corresponding to RG-flows and one-dimensional defects in the dual ABJM theory. Thus we take the metric given by a domain wall Ansatz ds 2 1,3 = e 2A(r) ds 2 1,2 +dr 2 , (3.15) and where the metric function, A(r), and the scalar fields, λ(r) and ζ(r), are functions of the radial coordinate, r, only. Furthermore, ds 2 1,2 , is either a Minkowski metric (RG-flows) or a metric on AdS 3 of radius ` (Janus solutions), ds 2 1,2 = e 2y/` (−dt 2 +dx 2 ) +dy 2 . (3.16) Since, at least formally, the equations for the RG-flows can be obtained by taking the radius `→∞, throughout much of the discussion we will write only the more general formulae for the Janus solutions. The equations of motion for the metric (3.15) and the scalar fields that follow from the Lagrangian (3.9) are λ 00 = − 3A 0 λ 0 + 1 4 sinh(4λ) (ζ 0 ) 2 − 2g 2 sinh(2λ), ζ 00 = − 3 A 0 ζ 0 − 4 coth(2λ)ζ 0 λ 0 , A 00 = − 3 2 (A 0 ) 2 − 3 2 (λ 0 ) 2 − 3 8 sinh 2 (2λ) (ζ 0 ) 2 − e −2A 2` 2 , (3.17) 27 and (A 0 ) 2 − (λ 0 ) 2 − 1 4 sinh(2λ) (ζ 0 ) 2 − 2g 2 cosh(2λ) + e −2A ` 2 = 0, (3.18) where last are two equations are independent combinations of the Einstein equations 2 . Imposing an unbroken supersymmetry along the flow, one obtains a first order system of the BPS equations. We refer the reader to [61] for further details and here only quote the final result: 3 λ 0 = − 1 3 A 0 W ! ∂W ∂λ + 2κ 3 e −A ` ! 1 sinh(2λ) 1 W ∂W ∂ζ , (3.19) ζ 0 = − 4 3 A 0 W ! 1 sinh 2 (2λ) ∂W ∂ζ − 2κ 3 e −A ` ! 1 sinh(2λ) 1 W ∂W ∂λ , (3.20) together with (A 0 ) 2 = g 2 W 2 − e −2A ` 2 . (3.21) The constant κ = ±1 is determined by the chirality of the unbroken supersymmetry, with N =(2, 0) for κ = 1 andN =(0, 2) for κ =−1. In the following we set κ = 1. Note that (3.21) is the same as (3.18) after one eliminates the derivatives of the scalar fields using (3.19) and (3.20). It is also straightforward to verify that the equations of motion (3.17) follow from the BPS equations. Finally, the BPS equations for supersymmetric RG-flows are A 0 = ±gW, (3.22) and λ 0 = ∓ g 3 ∂W ∂λ , ζ 0 = ∓ 4g 3 sinh 2 (2λ) ∂W ∂ζ . (3.23) These can be obtained from (3.19), (3.20) and (3.21) by taking the `→∞ limit. For GR flows 2 As a consequence of the Bianchi identities, the derivative of (3.18) follows from (3.17). 3 Similar BPS equations for holographic domain walls with curved slices were written down in [62–64]. 28 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 Figure 3.1: Typical flow trajectories for the Janus solutions to the BPS equations (3.19)–(3.21) in the (λ cosζ,λ sinζ)-plane. The background contours are of the superpotential W (λ,ζ). A red dot denotes the “central point” of a flow at (λ c cosζ c ,λ c sinζ c ) where A 0 = 0. there is no constraint on the chirality of the unbrokenN =2 supersymmetry. 3.2.3 Integrating the BPS equations The Janus solutions to the BPS equations (3.19)–(3.21) have been studied in [61] where it was shown by a numerical analysis that there are three classes of solutions shown in Figures 3.1 and 3.2: regular Janus solutions (shown in green) interpolating between two AdS 4 regions cor- responding to the same SO(8) stationary point of the potential (3.13) and singular solutions that diverge on either one side (shown in red) or both sides (shown in blue) of the flow. The Janus solutions are characterized by the presence of a special central point along a flow where the solution passes from one branch of (3.21) to another. This point is marked by a red dot and the corresponding values of the scalar fields are denoted by λ c andζ c , respectively. The position of this point for a given flow determines the type of a solution, see Figure 3 in [61]. In particular, for cosζ c 6=−1, all solutions are singular providedλ c is large enough. It is only when cosζ c =−1 that all solutions are regular Janus solutions irrespective of the value of λ c . In addition, there are solutions akin to RG-flows, which asymptote to AdS 4 on one side and 29 -6 -4 -2 2 4 6 r 1 2 3 4 5 AHrL -6 -4 -2 2 4 6 r 0.1 0.2 0.3 0.4 0.5 0.6 lHrL -6 -4 -2 2 4 6 r -3.5 -3.0 -2.5 -2.0 -1.5 -0.5 0.0 zHrL 1 2 3 4 r -4 -2 2 AHrL 1 2 3 4 r 0.5 1.0 1.5 lHrL 0 1 2 3 4 r 1.0 1.5 2.0 2.5 zHrL -0.6 -0.4 -0.2 0.2 0.4 0.6 r -1.6 -1.4 -1.2 -1.0 -0.8 AHrL -0.6 -0.4 -0.2 0.2 0.4 0.6 r 0.8 1.0 1.2 1.4 1.6 1.8 lHrL -0.6 -0.4 -0.2 0.2 0.4 0.6 r -0.6 -0.4 -0.2 0.2 0.4 0.6 zHrL Figure 3.2: Typical profiles of the metric function, A(r), and the scalar fields, λ(r) and ζ(r), for the different types of flows in Figure 3.1. become singular on the other, while remaining on a single branch of (3.21). They can be thought of as a singular limit of Janus solutions where the central point is moved off to infinity. Simplest examples of such flows are obtained by taking constant ζ = ζ 0 with cos(3ζ 0 )6=±1. Solving (3.19)–(3.21) for A 0 , λ 0 and A, and then imposing consistency between them, one is left with λ 0 = ∓ g √ 2 sinh(2λ) q cosh(2λ) + cos(3ζ 0 ) sinh(2λ), e −2A ` 2 = g 2 2 sin 2 (3ζ 0 ) sinh(2λ) cos(3ζ 0 ) + coth(2λ) . (3.24) Choosing the top sign in (3.24), we can impose the AdS 4 boundary condition in the UV, that is λ→ 0 asr→∞, to integrate the first equation for λ(r), and then solve the second equation for A(r). The resulting solutions are similar to the ones in Figure 3.4, which we will discuss shortly. The situation simplifies considerably in the RG-flow limit where the scalar equations (3.23) 30 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 Figure 3.3: RG-flow trajectories in the (λ cosζ,λ sinζ)-plane. The background contours are of the real superpotentialW (λ,ζ). The ridge trajectories have constantζ with cos 3ζ = 1 (green) and cos 3ζ =−1 (red), respectively. do not involve the metric function, A, and can be solved first. Choosing the top sign in (3.22)– (3.23), which corresponds to the UV region atr→∞, one then finds flows shown in Figures 3.3 and 3.4 [19]. In fact, as we have discussed in [19], generic solutions for the RG-flows can be determined analytically using two constants of motion: the general one I 1 = e 3A sinh 2 (2λ)ζ 0 , (3.25) valid for any ` and corresponding to the conserved current (3.10), and the second constant 4 I 2 = W 2 (cosh 2λ + cosζ sinh 2λ) 3 sin 3ζ sin 3 ζ = (4 cos 2 ζ− 1) 2 sin 2 ζ (3 + (cosh 2λ− 2 cosζ sinh 2λ) 2 ) (cosh 2λ + cosζ sinh 2λ) 2 , (3.26) for the first order system (3.22)–(3.23). UsingI 1 andI 2 , one can then determine A and ζ as a 4 Given different types of solutions in Figure 3.1, one would not expect to find such an additional constant of motion for the BPS equations (3.19)–(3.21) at finite `. 31 0 1 2 3 4 5 r-r 0 0 1 2 3 l HrL 2 4 6 8 r-r 0 1 2 3 4 AHrL Figure 3.4: Ridge flows for cos 3ζ =−1 (red) and cos 3ζ = 1 (green) with A 0 = 0. function ofλ, which is sufficient given the reparametrization invariance for the coordinate along the flow. However, this method of integration fails for the special flows with constant ζ = ζ 0 . For an RG-flow, one must then have cos 3ζ 0 =±1 (see, green and red ridge flows in Figure 3.3). The resulting equations can be obatained from (3.24) by taking the limit, cos(3ζ 0 ) −→ ±1, ` −→ ∞, `g sin(3ζ 0 ) −→ 2 √ 2e −A 0 , (3.27) where A 0 is a constant. One can then integrate those equations directly to obtain arccoth(e λ )± arctan(e λ )∓ π 2 = g √ 2 (r−r 0 ), (3.28) and A(r) = A 0 − log(e 4λ − 1) +          3λ for cos 3ζ = +1 λ for cos 3ζ =−1 (3.29) where r 0 and A 0 are integration constants. The solutions to those ridge flows are shown in Figure 3.4. Comparing with the trajectories in Figure 3.3, we expect that the solutions with cos(3ζ 0 ) = 1 are representative of the generic RG-flows, while those with cos(3ζ 0 ) =−1 are special since they lead to less singular four-dimensional metric at r = r 0 . This expectation is 32 confirmed by the asymptotic expansions that we will now discuss. 3.2.4 Behavior at large λ The holographic RG flows, governed by (3.22) and (3.23), have λ→∞ at some finite value of r. Generically, such solutions are dual to a massive flow toward some new infra-red limit. As was observed in [19], the RG flow solutions considered here can encode rich and interesting infra-red physics once one examines them in M theory. The Janus flows, governed by (3.19)– (3.21), can either form a loop starting and finishing at λ = 0 or start at λ = 0 and ultimately flow with λ→∞. There are also Janus solutions that begin and end with λ→∞. The Janus flows that involve largeλ may be viewed as interfaces that explore the infra-red structure of the holographic flow solutions. We will therefore examine the limiting behaviors of these flows as λ→∞. In Section 3.6 we will uplift these results to M theory to see more precisely how they may be interpreted in terms of M branes. From the explicit solutions in Figures 3.2 and 3.4, we see that the limit is characterized by λ −→ ∞, ζ −→ ζ ∞ , A −→ −∞ as r −→ r 0 , (3.30) where ζ ∞ is a constant asymptotic angle for a given flow. This observation is confirmed by a more careful expansion of the BPS equations (3.19)–(3.23). Expanding the superpotential (3.12) as λ→∞ for a generic ζ, we have W ∼ 1 4 q 1 + cos(3ζ)e 3λ +O(e −λ ), cos(3ζ) 6= −1, (3.31) while for the special ridge flows, W ∼ 3 2 √ 2 e λ +O(e −3λ ), cos(3ζ) = − 1. (3.32) 33 The flow equations (3.23) and (3.22) implify dA dλ = −3W ∂W ∂λ ! −1 ∼          −3 for cos 3ζ =−1 −1 otherwise . (3.33) dζ dλ = 4 sinh 2 (2λ) ∂W ∂ζ ∂W ∂λ ! −1 → 0. (3.34) The latter confirms the constancy ofζ at infinity while the former shows the rate of divergence of A depends upon that angle. This will translate into different physics once we uplift those flows to M theory. 3.3 The uplift We have obtained the Lagrangian (3.9) and the BPS equations (3.19)–(3.23) by a consistent truncation of the bosonic Lagrangian and the supersymmetry variations of four-dimensional, N = 8 gauged supergravity. Since the latter theory is a consistent truncation of M theory on S 7 [65–67], any solution of the equations of motion for the Lagrangian (3.9) can be uplifted to a solution of the eleven-dimensional supergravity. In the next two sections we verify this using explicit uplift formula for the metric [54] and the recently obtained uplift formulae for the flux [55–60]. Similar calculation verifying the new uplift Ansätze for the flux has been carried out recently for a special class of solutions of the four-dimensional,N = 8 theory given by some of the sta- tionary points of the potential: SO(8), SO(7) ± , G 2 , SU(4) − [56,58], and SO(3)×SO(3) [68], for which the four-dimensional space-time is AdS 4 and the scalar fields are constant. Our construc- tion of the uplift is similar as in those references, which the reader should consult for any omitted background material. 34 3.3.1 SU(3)×U(1)×U(1) invariants on S 7 The construction of an uplift inevitably leads to rather complicated formulae. Both to organize the calculation and to write down the result in a sucinct form, it is convenient to express the internal components of the fields in terms of canonical SO(7) tensors on S 7 that are associated with the E 7(7) generators of the scalar fields in the truncation. To this end let us define: 5 ξ mn = − 1 16 Φ + IJKL K IJ m K KL n , ξ m = 1 16 Φ + IJKL K IJ mn K nKL , ξ = ◦ g mn ξ mn , (3.35) and S mnp = 1 16 Φ − IJKL K IJ mn K KL p , (3.36) where Φ ± IJKL are the SO(8) tensors defined in (3.3) and K mIJ =i¯ η I ◦ Γ m η J , K IJ mn = ¯ η I ◦ Γ mn η J , ◦ D m K IJ n = m 7 K IJ mn , (3.37) are the SO(8) Killing vectors (one-forms) and two-forms on the round S 7 given in terms of an orthonormal basis of Killing spinors, η I , i ◦ D m η I = m 7 2 ◦ Γ m η I , ¯ η I η J = δ IJ , I,J = 1,..., 8. (3.38) The inverse radius of S 7 is denoted by m 7 ≡L −1 and Γ m = ◦ e a m Γ a . The circle indicates that ◦ e a is a siebenbein for the round metric on S 7 , ds 2 S 7 = ◦ g mn dy m dy n where ◦ g mn = ◦ e m a ◦ e n b δ ab , and ◦ D m is the covariant derivative with respect to that metric. Unless indicated otherwise, all indices on the S 7 tensors are raised and lowered with the round metric, for example K IJ m = ◦ g mn K nIJ . The coordinates, y m , on S 7 are for the moment arbitrary. However, one should note that ξ defined in (3.35) is a scalar harmonic on S 7 and may be thought of as providing a natural internal coordinate on the compactification manifold. By construction, the tensors (3.35)and (3.36) are invariant, i.e. havevanishing Lie derivative, 5 For a more extensive discussion of these tensors, see [68] and the original references therein. 35 under the SU(3)×U(1)×U(1)⊂ SO(8) symmetry group of the truncation. In particular, the Killing vectors for the two U(1)’s, υ m = Ω α IJ K IJ m , w m = Ω β IJ K IJ m , (3.39) Ω α 12 = Ω α 34 = Ω α 56 = 1, Ω β 78 = 1, (3.40) provide additional invariant one-forms on S 7 . In the following we will show that the metric and the flux for the uplift can be simply written in terms of the round metric, ◦ g mn , the one-forms ξ (1) ≡ ξ m dy m , υ (1) ≡ υ m dy m , ω (1) = ω m dy m , (3.41) the three-form, S (3) ≡ 1 6 S mnp dy m ∧dy n ∧dy p , (3.42) and the scalar, ξ. 3.3.2 The metric The eleven-dimensional space-time for the uplifted solutions is a warped product,M 1,3 ×M 7 , with the metric ds 2 11 = Δ −1 ds 2 1,3 +ds 2 7 , (3.43) where 6 ds 2 1,3 = ◦ g μν dx μ dx ν is the metric in four dimensions for a particular solution at hand. The internal metric, ds 2 7 = g mn dy m dy n , is determined by the celebrated formula for its densitized inverse [54]: Δ −1 g mn = 1 8 K mIJ K nKL u MN IJ +v MNIJ u MN KL +v MNKL , (3.44) 6 To distinguish the components of the four-dimensional metric (3.15) from the components of its eleven- dimensional counterpart along the four dimensions, we will denote the former by ◦ g μν . Thus g μν = Δ −1 ◦ g μν . 36 from which the warp factor, Δ, can be calculated using Δ −9 = det(Δ −1 g mn ◦ g np ). (3.45) Whileitispossibletoexpressthedensitizedmetricentirelyusingtensors(3.35)and(3.36)and their (contracted) products, 7 the simplest expression is obtained by noting that the symmetric tensors resulting from such contractions can be rewritten using the round metric, ◦ g mn , and bilinears in the one forms ξ m , υ m and ω m , as in the following examples: ξ mn = 1 6 (3 +ξ) ◦ g mn + 1 6(ξ− 3) ξ m ξ n + 3 8(ξ− 3) (υ m +ω m )(υ n +ω n ), S mpq S n pq = 1 4 (υ m −ω m )(υ n −ω n ), S mpr S nq r ξ p ξ q = 9 4 υ m υ n + 3 4 (9− 2ξ)ω m ω n + 3ξ 2 υ (m ω n) . (3.46) After some algebra, we then find Δ −1 g mn = c 1 ◦ g mn +c 2 ξ m ξ n +c 3 υ m υ n +c 4 ω m ω n +c 5 υ (m ω n) , (3.47) where all the c i can be expressed in terms of four-dimensional quantities and the scalar, ξ: c 1 = cosh(2λ)− 1 6 (ξ + 3) sinh(2λ) cos(ζ), c 2 = 1 6(3−ξ) sinh(2λ) cos(ζ), c 3 = sinh(2λ) 32(ξ− 3) (ξ− 3) cosh(4λ) cos(ζ) + (ξ− 3) sinh(4λ)− (ξ + 9) cosζ , c 4 = sinh(2λ) 16(ξ− 3) (ξ− 3) sinh 2 (2λ) cos(3ζ) + 1 2 (ξ− 3) sinh(4λ)− 6 cosζ , c 5 = sinh(2λ) 16(ξ− 3) (ξ− 3) sinh(4λ) cos(2ζ) + (ξ− 3) cosh(4λ) cosζ− (ξ + 9) cosζ . (3.48) 7 See, for example a general discussion in [68]. 37 Note that the indices on the right hand side in (3.46) are raised using the round metric, ◦ g mn , which is the convention followed throughout this section. All that is needed now to invert the densitized metric (3.47) are contraction identities between the one-forms, which can be derived using the explicit form of the SO(8) tensors and properties of the Killing vectors summarized in [56,68] and the references therein. We have ξ m ξ m = 27−6ξ−ξ 2 , υ m υ m = 12− 8 3 ξ, ω m ω m = 4, ξ m υ m = ξ m ω m = 0, υ m ω m = − 4 3 ξ. (3.49) It is then straightforward to check that Δg mn = g 1 ◦ g mn +g 2 ξ m ξ n +g 3 (υ m υ n +ω m ω n ) +g 4 υ (m ω n) , (3.50) where g 1 = 6 6 cosh(2λ)− (ξ + 3) sinh(2λ) cosζ , g 2 = D 36(ξ− 3) cosζ h 3 sinh(4λ)− (ξ + 3) sinh 2 (2λ) cosζ i , g 3 = D 16(ξ− 3) h 3 sinh(4λ) cos(ζ)− sinh 2 (2λ)(ξ + 3 cos(2ζ)) i , g 4 = D 8(ξ− 3) h 3 sinh(4λ) cos(ζ)− sinh 2 (2λ)(ξ cos(2ζ) + 3) i , (3.51) and D = 36 h sinh(2λ) cosζ + cosh(2λ) ih 6 cosh(2λ)− (ξ + 3) sinh(2λ) cosζ i 2 . (3.52) Using the contractions (3.49), one can also calculate the derivatives of the warp factor given by (3.45) with respect to λ and ζ. Then a simple integration yields Δ =D 1/3 , (3.53) 38 with the overall normalization set by Δ = 1 for the round sphere metric when λ = 0. Dividing out Δ in (3.50), yields the internal metric, g mn , in terms of the SO(7) tensors associated with the truncation. 3.3.3 Internal coordinates and local expressions In addition to the metric tensor, we will also need the corresponding orthonormal frames and those turn out to be rather cumbersome to write down using the invariant tensors (3.50). Also, theformulaelike(3.50)tendtoobscuretheunderlyinggeometryofthesolutionanditssymmetry. To address both of these points, we will now choose a suitable set of coordinates, y m , on the internal manifold. As usual, see for example [68] section 7.1, this can be done systematically as follows: First, we embed S 7 into the ambientR 8 as the surface Y A Y A = m −1 7 , (3.54) such that the Killing vectors K IJ = K IJ m dy m defined in (3.37) are related by triality to the familiar ones, that is K IJ = − m 7 2 Γ IJ AB K AB , K AB = − 1 8m 7 Γ IJ AB K IJ , (3.55) where K AB = Y A dY B −Y B dY A , (3.56) Similarly, we have K IJ (2) ≡ 1 2 K IJ mn dy m ∧dy n = 1 2 Γ IJ AB dK AB . (3.57) The action of the symmetry group SU(3)×U(1)×U(1) in the ambient space is given by the 39 branching 8 8 s → (3,− 1 2 , 1 2 ) + ( ¯ 3, 1 2 ,− 1 2 ) + (1, 3 2 , 1 2 ) + (1,− 3 2 ,− 1 2 ). (3.58) Onecanchoosearepresentationof Γ-matricessuchthatthe SU(3)generatorsactinthesubspace, Y 1 ,...,Y 6 , while the two U(1) generators have 2× 2 diagonal blocks. Then a convenient choice for the coordinates, (y m ) = (χ,θ,α 1 ,α 2 ,α 3 ,ψ,φ), on S 7 , that makes the symmetry manifest is as follows: Y 1 +iY 2 = m −1 7 cosχ sinθ sin α 1 2 e i 2 (α 2 −α 3 ) e −i(φ+ψ) , Y 3 +iY 4 = m −1 7 cosχ sinθ cos α 1 2 e − i 2 (α 2 +α 3 ) e −i(φ+ψ) , Y 5 +iY 6 = m −1 7 cosχ cosθe −i(φ+ψ) , Y 7 +iY 8 = m −1 7 sinχe −iφ , (3.59) whereα 1 ,α 2 ,α 3 are the SU(2) Euler angles, while the anglesψ andφ parametrize the U(1)×U(1) isometry. 9 In this parametrization, the round metric on S 7 with unit radius is 10 ds 2 S 7 ≡ m 2 7 dY A dY A = dχ 2 + cos 2 χ ds 2 CP 2 + sin 2 χ dψ + 1 2 sin 2 θσ 3 2 + dφ + cos 2 χ dψ + 1 2 sin 2 θσ 3 2 , (3.60) where ds 2 CP 2 = dθ 2 + 1 4 sin 2 θ σ 2 1 +σ 2 2 + cos 2 θσ 2 3 ), (3.61) is the metric onCP 2 and σ i are the SU(2)-invariant forms. The first line in (3.60) is the metric onCP 3 and the second line is the Hopf fiber. The SU(3)× U(1) ψ symmetry acts transitively on CP 2 and the ψ-fiber. Both the metric onCP 2 and the one form dψ + 1 2 sin 2 θ are invariant. 8 Wefollowheretheusualconventionthatthesupersymmetries, i , transformin8 v , whiletheambientcoordinates, Y A , in 8 s of SO(8). 9 More precisely, the two U(1) angles are φ +ψ/2 and−φ− 3ψ/2, respectively. 10 All functions and forms in the ambientR 8 are implicitly pulled-back onto S 7 using (3.54). 40 Using (3.55) and (3.59), we can now express the invariants introduced in section 3.3.1 in terms of ambient and local coordinates. We find that the invariant function, ξ, is simply ξ = 3− 12m 2 7 h (Y 7 ) 2 + (Y 8 ) 2 i = − 9 + 12m 2 7 h (Y 1 ) 2 +... + (Y 6 ) 2 i = 3(1− 4 sin 2 χ), (3.62) while the invariant one-forms are ξ (1) = − 12m 7 (Y 7 dY 7 +Y 8 dY 8 ) = − 6m −1 7 sin(2χ)dχ, υ (1) +ω (1) = 8m 7 (Y 7 dY 8 −Y 8 dY 7 ) = − 8m −1 7 sin 2 χdφ, υ (1) − 3ω (1) = 8m 7 (Y 1 dY 2 −Y 2 dY 1 +... +Y 5 dY 6 −Y 6 dY 5 ) = − 8m −1 7 cos 2 χ (dφ +dψ + 1 2 sin 2 θσ 3 ). (3.63) We also have S (3) ≡ 1 6 S mnp dy m ∧dy n ∧dy p = − 1 6 m 7 Φ + MNPQ Y M dY N ∧dY P ∧dY Q = −m −3 7 J CP 3∧ϑ S 7, (3.64) where J CP 3 = 1 2 dϑ S 7, ϑ S 7 = dφ + cos 2 χ (dψ + 1 2 sin 2 θσ 3 ), (3.65) are, respectively, the complex structure onCP 3 and the corresponding Sasaki-Einstein one-form on S 7 . Finally, we substitute the invariants (3.62) and (3.63) into the warp factor (3.53) and the metric (3.50). To simplify expressions we define: X ± (x) ≡ cosh 2λ± cosζ sinh 2λ, Σ(x,χ) ≡ X + sin 2 χ +X − cos 2 χ, (3.66) which are functions of the space-time coordinates, x μ , and the internal coordinate, χ. Then the 41 internal metric can be written as ds 2 7 = m −2 7 Σ X 2 3 dχ 2 + cos 2 χ X Σ ds 2 CP 2 + sin 2 χ X Σ (dψ + 1 2 sin 2 θσ 3 + Ξ X dφ) 2 + 1 Σ 2 dφ + cos 2 χ dψ + 1 2 sin 2 θσ 3 2 , (3.67) where to simplify the notation we set X≡X + and Ξ≡X + −X − . The warp factor (3.53) is Δ = 1 X 1/3 Σ 2/3 . (3.68) For λ = 0, we have X ± = Σ = 1 and the metric (3.67) reduces to the metric (3.60) on the sphere with radius m −1 7 . The deformation clearly preserves the SU(3)×U(1)×U(1) symmetry as well as the metric along the Hopf fiber, which is now rescaled by Σ −2 with respect to the six-dimensional base. This suggests that there might be some deformed Kähler geometry still present in the background. We will return to this point below in section 3.6. 3.3.4 The transverse flux It is rather remarkable that it took more than 25 years to obtain workable formulae for the four-form flux, F (4) = dA (3) . Indeed, while the general proof of the consistent truncation of eleven-dimensional supergravity on S 7 [65–67] yielded explicit formulae for F (4) , those formulae wereratherdifficultifnotimpossibletouseforallbutthesimpleststationarypointsolutions[67]. It is only recently that new Ansätze for various components of the four-form flux were found in [55–60] whose complexity is comparable to that of the metric Ansatz. Starting with a domain wall solution in four-dimensions with a metric as in (3.15) and scalar fields depending only on the transverse coordinate, the corresponding four-form flux in eleven- dimensional supergravity can be decomposed into a sum of two terms F (4) = F st (4) +F tr (4) , (3.69) 42 where F st (4) = F (4,0) +F (3,1) is the “space-time” flux and F tr (4) = F (0,4) +F (1,3) is the “transverse” flux. A label (4−p,p) indicates a (4−p) th order form alongM 1,3 and ap th order form along the internalmanifold,M 7 . SincebythePoincaréorconformalsymmetryalongthethree-dimensional slices inM 1,3 there can be no (2, 2)-form in (3.69), 11 the Bianchi identity,dF (4) = 0, implies that both F st (4) and F tr (4) must be closed. Hence F tr (4) = dA tr (3) , where A tr (3) can have at most one “leg” along dr and thus can be always gauge transformed into a 3-form with all three legs along the internal manifoldM 7 , that is A tr (3) = 1 6 A mnp dy m ∧dy n ∧dy p . The componentsA mnp are given by the new uplift Ansatz [55,56], which, in our conventions, reads Δ −1 g pq A mnp = i 16 K IJ mn K qKL u MN IJ −v MNIJ u MN KL +v MNKL . (3.70) It is convenient to define a two-form S m ≡ 1 2 S mnp dy n ∧ dy p . Evaluating (3.70) in terms of invariants, we then find 1 2 Δ −1 g pq A mnp dy m ∧dy n = (a 11 υ q +a 12 ω q )dυ + (a 21 υ q +a 22 ω q )dω +a 3 S q , (3.71) where the vector index on the right hand side is raised using the round metric and the coefficients are given by a 11 = 1 64 m −1 7 sinh 3 (2λ) sinζ, a 12 = 1 64 m −1 7 sinh 2 (2λ) sinζ h 2 cosh(2λ) cosζ− sinh(2λ) i , a 21 = − 1 64 m −1 7 sinh 2 (2λ) sinζ h 2 cosh(2λ) cosζ + sinh(2λ) i , a 22 = − 1 64 m −1 7 sin(3ζ) sinh 3 (2λ), a 3 = − 1 2 sinζ sinh(2λ). (3.72) Contracting with the densitized metric (3.50) and then using the contraction identities (3.49) 11 Such terms must also vanish whenever the vector fields in four dimensions are set to zero [60]. 43 together with ξ m S m = − 3 4 υ (1) ∧ω (1) , υ m S m = 1 12 m −1 7 (ξ− 6)dυ (1) + 1 4 m −1 7 dω (1) − 1 6 ξ (1) ∧υ (1) , ω m S m = 1 12 m −1 7 ξdυ (1) − 1 4 m −1 7 dω (1) − 1 6 ξ (1) ∧υ (1) , (3.73) we find that the internal potential is simply given by A tr (3) = α 1 S (3) +α 2 ξ (1) ∧υ (1) ∧ω (1) , (3.74) where α 1 = − 1 2 Σ sinζ sinh(2λ), α 2 = − 1 384 X Σ sin(2ζ) sinh 2 (2λ) sin 2 χ . (3.75) Rewriting (3.74) in local coordinates using (3.63) and (3.64) yields A tr (3) = 1 2 m −3 7 sinζ sinh(2λ) Σ J CP 3− 1 2 sin(2χ) Ξ X dχ∧dφ ∧ϑ S 7. (3.76) Note that the A tr (3) has only components along the internal manifold,M 7 , so that its field strength, F tr (4) , can have at most one leg along the four-dimensional space-time. 3.3.5 The space-time flux We now turn to the second part of the flux, F st (4) , which, as shown recently in [57,58], can be determined from the uplift for the transverse dual potential, A tr (6) . The starting point is the Maxwell equation (A.3) in eleven dimensions, which by setting 44 F (4) =dA (3) can be written locally as 12 d(?F (4) +A (3) ∧F (4) ) = 0, (3.77) from which the dual potential, A (6) , is defined by dA (6) = ? F (4) +A (3) ∧F (4) . (3.78) The space time flux, F st (4) , is determined by the transverse part of A (6) , that is F st (4) = −? dA tr (6) −A tr (3) ∧F tr (4) , (3.79) where A tr (6) = 1 16 T (6) − 3m 7 ◦ ζ (6) . (3.80) The six-form, T (6) = 1 6! T m 1 ...m 6 dy m 1 ∧...∧dy m 6 , is given by the uplift Ansatz T m 1 ...m 6 = Δg p[m 1 K m 2 ...m 6 ] IJ K pKL (u MN IJ +v MNIJ )(u MN KL +v MNKL ), (3.81) where K IJ m 1 ...m 5 ≡i ¯ η I Γ m 1 ...m 5 η J , while ◦ ζ (6) is the potential for the volume of the round S 7 , d ◦ ζ (6) ≡ v ◦ ol S 7 = 1 8 m −7 7 sinχ cos 5 χ sin 3 θ cosθdχ∧dθ∧σ 1 ∧σ 2 ∧σ 3 ∧dψ∧dφ. (3.82) Evaluating (3.81), we find T (6) = 8 3 +ξ− 6 coth(2λ) secζ ◦ ∗ 7 ξ (1) , (3.83) 12 See appendix A for definitions and properties of the various duals used in this section. 45 where ◦ ∗ 7 is the dual on S 7 with respect to the round metric. In terms of the local coordinates, T (6) = 2m −6 7 sin 2 χ cos 6 χ sin 3 θ cosθ cos(2χ)− coth(2λ) secζ dθ∧σ 1 ∧σ 2 ∧σ 3 ∧dψ∧dφ. (3.84) Then dT (6) =− 4m −1 7 csch 2 (2λ) sec 2 ζ sin(2χ) (coth(2λ) secζ− cos(2χ)) 2 h 4 cosζdλ− sinζ sinh(4λ)dζ i ∧ı ∂χ v ◦ ol S 7 + 8m 7 4 coth(2λ) secζ(1− 2 cos(2χ))− 4 cos(2χ) + 3 cos(4χ) + 5 (coth(2λ) secζ− cos(2χ)) 2 v ◦ ol S 7. (3.85) From (3.76), we have A tr (3) ∧F tr (4) = −m 7 sin 2 ζ h (1− 2 cos(2χ)) cosζ + 3 coth(2λ) i (cosζ + coth(2λ))(coth(2λ)− cosζ cos(2χ)) 2 v ◦ ol S 7. (3.86) Substituting (3.85), (3.82) and (3.86) in (3.79), we get F st (4) =? m −1 7 sin(2χ) (cosζ sinh(2λ) cos(2χ)− cosh(2λ)) 2 cosζdλ− 1 4 sinζ sinh(4λ)dζ ∧ı ∂χ v ◦ ol S 7 − m 7 XΣ 2 cosζ sinh(2λ)(2 cos(2χ)− 1)− 3 cosh(2λ) v ◦ ol S 7 , (3.87) where the dual is with respect to the full metric (3.43). Using identities (A.7) and (A.8) in appendix A and ? v ◦ ol S 7 = Δ −1 ? vol M 7 = − Δ −3 v ◦ ol 1,3 , (3.88) we find that the space time flux (3.87) is F st (4) = −m −1 7 sin(2χ) cosζ ◦ ∗ 1,3 dλ− 1 4 sinζ sinh(4λ) ◦ ∗ 1,3 dζ ∧dχ + m 7 Δ −3 XΣ 2 cosζ sinh(2λ)(2 cos(2χ)− 1)− 3 cosh(2λ) v ◦ ol 1,3 . (3.89) For the flow solutions where the scalar fields depend only on the radial coordinate,r, we have 46 dλ =λ 0 dr, dζ =ζ 0 dr and (3.89) evaluates to a very simple expression, F st (4) = m 7 3 e 3A v ◦ ol 1,2 ∧ (Udr +V dχ), (3.90) where U(r,χ) = −3(1− 2 cos 2χ) sinh 2λ cosζ− 9 cosh 2λ, V (r,χ) = 3 4m 2 7 sin 2χ (4 cosζλ 0 − sinh(4λ) sinζζ 0 ), (3.91) and v ◦ ol 1,2 is the volume along the Min 1,2 or AdS 3 slices. ItisstraightforwardtoverifythatF st (4) givenin(3.105)satisfiestheBianchiidentity,dF st (4) = 0, when the four-dimensional fields, A(r), λ(r) and ζ(r), are on-shell, that is they satisfy the equations of motion (3.17) in four dimensions. The calculation above illustrates the point we have raised before, namely, that a rather long and complicated derivation using uplift formulae yields a relatively simple final result. In fact, after we have completed this calculation a paper [60] appeared where a more direct Ansatz for the Freund-Rubin flux, namely the term in F st (4) proportional to the volume of the four-dimensional space-time, is proposed. In the present context, the key observation is that the second term in U in (3.91) is the scalar potential (3.13) of the four-dimensional theory, while the first term is proportional to a derivative of the potential. This can be generalized to a more efficient uplift formula, which is summarized in appendix C. 3.3.6 A summary of the uplift We conclude this section with a brief summary of the eleven-dimensional fields constructed in sections 3.3.2–3.3.5. While the formulae for the uplifted fields are valid for any field configuration in four dimensions, here we will specialize them to the four-dimensional flows we are interested in. It turns out that the simplest form of the flux is obtained when we use suitable frames for the metric (3.43). We will also need those frames later in the proof of supersymmetry of the RG 47 flows and Janus solutions in section 3.5. Given (3.43) and (3.67), a natural choice for the frames, e M , M = 1,..., 11, is to set e 1,2,3 = X 1 6 Σ 1 3 e A f 1,2,3 , e 4 = X 1 6 Σ 1 3 dr, e 5 = m 7 X − 1 3 Σ 1 3 dχ, e 6 = m 7 X 1 6 Σ − 1 6 cosχdθ, e 7,8 = m 7 2 X 1 6 Σ − 1 6 cosχ sinθσ 1,2 , e 9 = m 7 2 X 1 6 Σ − 1 6 cosχ sinθ cosθσ 3 , e 10 = m 7 X 2 3 Σ − 2 3 sinχ cosχ (dψ + 1 2 sin 2 θσ 3 ) + Ξ X dφ , e 11 = m 7 X − 1 3 Σ − 2 3 dφ + cos 2 χ (dψ + 1 2 sin 2 θσ 3 ) , (3.92) where f i , i = 1, 2, 3, are the frames for the Min 1,2 or AdS 3 slices, X(r) = cosh(2λ) + cosζ sinh(2λ), Ξ(r) = 2 cosζ sinh(2λ), Σ(r,χ) = cosh(2λ)− cosζ sinh(2λ) cos(2χ). (3.93) Then the transverse potential, A tr (3) , given in (3.76) becomes surprisingly simple, A tr (3) = 1 2 p(r) (e 6 ∧e 9 +e 7 ∧e 8 −e 5 ∧e 10 )∧e 11 , (3.94) where p(r) = sinh(2λ) sinζ. (3.95) Note that the coefficient function, p(r), depends only on the four-dimensional space time radial coordinate. All dependence in (3.94) on the internal geometry and coordinates enters only through the frames. Finally, the space time flux is given in (3.105) and (3.91). This completes the constriction of 48 the uplift. 3.4 The equations of motion In this section we verify explicitly that the metric and the four-form flux in the uplift satisfy the equations of motion of eleven-dimensional supergravity when the four-dimensional metric and the scalar fields satisfy the four-dimensional equations of motion (3.17)–(3.18). 3.4.1 Preliminaries We start with some technical preliminaries that will help us simplify the algebra in the calcu- lations that follow. The main idea is to work directly with the functions that appear in the metric (3.67) and the flux (3.94), in particular, with X(r) and p(r) given in (3.93) and (3.95), respectively, rather than with the scalar fields, λ(r) and ζ(r). To this end we use sinζ = −p csch(2λ), cosζ = − csch(2λ)(cosh(2λ)−X), (3.96) and cosh(2λ) = 1 +p 2 +X 2 2X , (3.97) to eliminate ζ and λ in terms of p and X. This converts complicated trigonometric expressions into rational functions of the new fieldsp andX that are typically easier to evaluate and simplify. In particular, the four-dimensional equations of motion (3.17)–(3.18) in the rationalized form are 49 given by Δ 00 = − 3A 0 Δ 0 + 1 X p 2 + 1 (X 0 ) 2 +X 2 (p 0 ) 2 − 2pXp 0 X 0 − 2g 2 p 2 +X 2 − 1 p 00 = − 3p 0 A 0 + p X 2 p 2 + 1 (X 0 ) 2 +X 2 (p 0 ) 2 − 2pXp 0 X 0 − 2g 2 p X p 2 +X 2 + 1 , A 00 = − 3 2 (A 0 ) 2 − 3 8X 2 p 2 + 1 (X 0 ) 2 +X 2 (p 0 ) 2 − 2pXp 0 X 0 + 3g 2 2X 1 + (p 0 ) 2 + (X 0 ) 2 − e −2A 2` 2 , (3.98) and (A 0 ) 2 − 1 4X 2 p 2 + 1 (X 0 ) 2 +X 2 (p 0 ) 2 − 2pXp 0 X 0 − g 2 X p 2 +X 2 + 1 + e −2A ` 2 = 0. (3.99) Similarly, we find that the superpotential (3.12) is given by W 2 = 1 8X 9p 4 − 6p 2 X 2 − 3 + X 2 + 3 2 (3.100) and the BPS equations (3.19) and (3.20) for the scalars become X 0 = − 1 4W 2 h 9p 4 − 6p 2 X 2 − 1 +X 4 + 2X 2 − 3 i A 0 + e −A ` p (3p 2 −X 2 + 3) W 2 , p 0 = − p 4XW 2 h 9p 4 − 6p 2 X 2 − 3 +X 4 − 2X 2 + 9 i A 0 + e −A ` 3p 4 + 2p 2 (X 2 + 3)−X 4 − 2X 2 + 3 4XW 2 . (3.101) As a consistency check one can verify once more that the first order equations (3.101) and (3.21) withW given in (3.100) imply the second order equations (3.98) and that (3.99) is equivalent to (3.21). 50 Finally, the other metric and the flux functions are: Σ = 1 X h cos 2 χ(1 +p 2 ) + sin 2 χX i , Ξ = − 1 X p 2 −X 2 + 1 , (3.102) and U = − 6 X cos 2 χ (1 +p 2 ) + 3X(cos(2χ)− 2), V = 3 4m 2 7 sin 2χ 2(1 +p 2 ) X 0 X − 2pp 0 . (3.103) This shows that indeed both the metric and the flux can almost entirely be written down using, up to overall factors, only rational functions of X and p, and their derivatives! Finally, we will be often able to eliminate trigonometric functions of χ using cos(2χ) = − p 2 +X 2 − 2 ΣX + 1 p 2 −X 2 + 1 , (3.104) which follows from (3.102). 3.4.2 The flux The first place where using the rationalized parametrization becomes clearly advantageous is the calculation of the components, F MNPQ , of the four-form flux, F (4) . Indeed, for the space-time part of the flux given in (3.105) we simply have F 1234 = m −1 7 Σ 4/3 X 2/3 (2 Σ +X), F 1235 = tanχ X 7/6 Σ 4/3 Ξ (Σ−X) p 2 + 1 X 0 −pXp 0 . (3.105) Turning to the transverse flux, F tr (4) = dA tr (3) , we note that the part of the three-form poten- tial alongCP 2 in (3.76) has the complex structure, J CP 2, as a factor. Thus the corresponding 51 components of the field stength must satisfy F MN69 = F MN78 . (3.106) Modulo this identity, the non-vanishing components of the transverse part of the flux are: F 4510 11 = pX 0 −Xp 0 2X 7/6 Σ 1/3 , F 469 11 = Σp 0 −pΣ 0 2X 1/6 Σ 4/3 , F 569 10 = − m 7 p(Σ +X) X 2/3 Σ 4/3 , F 6789 = 2m 7 p X 2/3 Σ 1/3 , (3.107) where Σ 0 ≡ ∂Σ ∂r = 1 X 3 − (p 2 + 1)X 2X pXp 0 − (p 2 + 1)X 0 + Σ p 2 X 0 − 2pXp 0 + X 2 + 1 X 0 . (3.108) Later we will also need ∂Σ ∂χ = sin(2χ) Ξ. (3.109) It appears that the flux produced through the uplift formulae is rather special, in particular, we find that the following components F 4569 = F 469 10 = F 569 11 = 0, (3.110) accidentally vanish, that is not due to the underlying SU(3)×U(1)×U(1) symmetry of the con- struction. 52 3.4.3 The Einstein equations The Einstein equations of eleven-dimensional supergravity in our conventions 13 are: R MN +g MN R = 1 3 F MPQR F N PQR . (3.111) We start by evaluating the components of the Ricci tensor, R MN , in the basis of frames (3.92). The symmetries of the metric in four dimensions and the dependence of the scalar fields on the radial coordinate only, imply that the non-vanisnhing components of the Ricci tensor can be at most the following ones: R 11 = −R 22 = −R 33 , R 44 , R 45 = R 54 , R 55 , R 66 = R 77 = R 88 = R 99 , R 10 10 , R 10 11 = R 11 10 , R 11 11 . (3.112) This agrees with the explicit result. Indeed, we find that after imposing the four-dimensional equations of motion (3.98)–(3.99) in the rational parametrization introduced above, the diagonal components of the Ricci tensor can be written in the form R MM =A M (X 0 ) 2 +B M (p 0 ) 2 +C M p 0 X 0 +D M , (3.113) whereA,B,C andD are functions of p, X and χ (or, equivalently, Σ). In particular, we find that the cross-termsA 0 X 0 andA 0 p 0 are absent. Similarly, the off-diagonal components are of the form R MN =A MN X 0 +B MN p 0 +D MN , M6=N. (3.114) Explicit formulae for all non-vanishing coefficient functions are given in appendix D. Evaluating the energy-momentum tensor on the right hand side in (A.1) is straightforward. Wewillforegothedetailsandjustlookatonespecificequation, theoff-diagonalEinsteinequation 13 See appendix A. 53 (A.1) with M = 10 and N = 11. On the one side, we have R 10 11 = − 2 g 2 − 2m 2 7 tanχ (Σ−X) X 1/3 Σ 5/3 . (3.115) However, as one can see by inspection of the non-vanishing flux components, the other side must be zero. This verifies the relation between the four-dimensional coupling constant, g, and the inverse radius of the internal manifold, m 7 , [66] g = √ 2m 7 . (3.116) Given this relation, it is easy to check that all the remaining Einstein equations are satisfied as expected. 3.4.4 The Maxwell equations The Maxwell equations are d?F (4) +F (4) ∧F (4) = 0. (3.117) For the flux (3.105)–(3.107), they yield seven independent equations: four first order and three second order. The first order equations are along the components [1234569 11], [1234578 11], [12356789], and [456789 10 11], and all have the same structure as this last one 4m 7 p tanχ g 2 X 17/6 Σ 5/3 Ξ 2 2m 2 7 −g 2 p 2 −X 2 + 1 (Σ−X) p 2 X 0 −pXp 0 +X 0 = 0, (3.118) namely, they come with an overall factor of (g 2 − 2m 2 7 ). Thesecondorderequationscomefromthecomponents [1234569 10], [1234578 10]and [12346789] in (A.3). The first two equations are somewhat involved, but the last one is quite simple. There 54 we find 0 = 1 2X 4/3 Σ 2/3 pX 00 −Xp 00 + 3A 0 (pX 0 −Xp 0 )− 24m 2 7 = 2p (g 2 − 2m 2 7 ) X 4/3 Σ 2/3 , (3.119) where in going from the first to the second line we have used the four-dimensional equations of motion (3.98). Similarly, upon using (3.98), the other two equations reduce to the same expression modulo an overall factor of XΣ. Thus the Maxwell equations are satisfied if (3.116) holds. To summarize, we have shown explicitly that the metric, g MN , and the four-form flux, F (4) , constructed using the uplift formulae in section 3.3 indeed satisfy the equations of motion of the eleven-dimensional supergravity when the scalar fields, λ(r) and ζ(r), and the metric function, A(r), are on-shell in four dimensions. It is important to note that to verify that we have used only the equations of motion in four dimensions (3.17)–(3.18) or, equivalently, (3.98)–(3.99), but not the BPS equations! This means that also non-supersymmetric solutions of the same type will uplift to solutions of M theory. 3.5 Supersymmetry We now turn to the Janus and RG-flow solutions of the BPS equations (3.19)–(3.21) and (3.22)– (3.23), respectively, to demonstrate explicitly theN = (0, 2) andN = 2 supersymmetry of the corresponding uplifts in M theory. This has been discussed already in some detail in [19], where we have argued that theN =2 supersymmetry of the RG flows is achieved by brane polarization and is naturally defined through projectors that reflect the underlying almost-complex structure and a dielectric projector much like those encountered in [69–72]. The defect in the Janus solutions leads to additional chiral projector that is also present in four dimensions. The result for the RG-flows is then recovered by keeping both chiralities and taking the `→∞ limit. 55 3.5.1 Projector Ansätze The BPS equations in eleven dimensions are obtained by setting the supersymmetry variations of the gravitinos to zero, δψ M ≡ ∂ M +M M = 0, (3.120) where the algebraic operators,M M , are given by M M ≡ 1 4 ω MPQ Γ PQ + 1 144 Γ M NPQR − 8δ M N Γ PQR F MNPQ . (3.121) The Killing spinors of unbroken supersymmetries are invariant under the Poincaré transforma- tions in thetx-plane and are singlets of SU(3) acting alongCP 2 . Hence does not depend on the cordinatest,x andθ, as well as the Euler angles,α 1 ,...,α 3 . This means that the corresponding equations (3.120) are purely algebraic: 14 M t =M x = 0, M θ =M σ 1 = ... =M σ 3 = 0. (3.122) Similarly, the dependence of on the U(1)× U(1) angles, φ and ψ, ∂ ∂φ = −M φ , ∂ ∂ψ = −M ψ , (3.123) is determined by the charges, q φ = 1 and q ψ = 3/2, respectively. Let us now consider the first equation in (3.122), written in the form M = 0, M ≡ Γ 1 M 1 . (3.124) 14 We will use the convention that the indices M = 1,..., 11 label components with respect to the frames (3.92), while M =t,x,...,ψ, or M =σ 1 ,...,σ 3 , with respect to the local coordinates and/or forms. 56 The matrixM, expanded into the basis of Γ-matrices, is given by M = e −A 2` 1 X 1/6 Σ 1/3 Γ 3 + 1 12X 7/6 Σ 4/3 2X ∂Σ ∂r + Σ (X 0 + 6XA 0 ) i Γ 4 + m 7 X 1/3 6Σ 4/3 ∂Σ ∂χ Γ 5 + 1 3 F 1234 Γ 4 +F 1235 Γ 5 Γ 123 + 1 6 F 45 10 11 Γ 45 10 11 + 1 6 F 6789 Γ 6789 + 1 6 F 469 11 Γ 4 11 +F 569 10 Γ 5 10 Γ 69 + Γ 78 . (3.125) Together with the explicit formulae (3.105)–(3.107) for the flux, this gives us a homogenous system of linear equation for the thirty two components of . It is clear that after substituting the expressions for the flux components (3.105)–(3.107) and expanding the derivatives of Σ, see (3.108) and (3.109), the operator M, as well as the other operators,M M , become quite complicated. Hence, before we proceed with the analytic calculation, we first explore numerically the space of solutions to (3.124). To do that, we first eliminate the derivatives X 0 , p 0 and A 0 using the BPS equations (3.101) and (3.21), and set g = √ 2m 7 . NextweassignrandomvaluestothefieldsX,p,A, theangleχ, andtheconstantsm 7 and ` upon which (3.124) becomes a purely numerical system that can be solved for the components of the Killing spinor,. Note that our numerical assignment amounts simply to choosing random initial conditions for the four-dimensional BPS equations and thus is not constrained in any way. Those numerical solutions yield us some information about the subspace of allowed Killing spinors, which confirms what one could also infer from an analysis in four dimensions and the SU(3)×U(1)×U(1) symmetry. More importantly, it allows us to short cut quite a bit of tedious analysis by fixing some of the signs in the projectors below that we would have to keep track of otherwise. For finite`, the space of numerical solutions is generically two-dimensional in agreement with N = (2, 0) supersymmetry in four dimensions. The unbroken supersymmetries, , must thus satisfy four conditions Π 0 = Π 1 = Π 2 = Π 3 = 0, (3.126) 57 where Π 0 ,..., Π 3 are mutually commuting projectors. From the numerical analysis we also find that two of these projectors are constant. To conform with the conventions in [19], we will denote them by Π 1 and Π 3 . The first projector, Π 1 ≡ 1 2 (1 + Γ 6789 ), (3.127) arises from the fact that the Killing spinor, , must be a singlet under the holonomy group, SU(3), ofCP 2 . It depends on the choice of orientation ofCP 2 defined by the frames e 6 ,...,e 9 . The second projector, Π 3 ≡ 1 2 (1− Γ 12 ), (3.128) isjustanupliftofthecorrespondingchiralityprojectorinfourdimensions. Inparticular, choosing κ =−1 in (3.19) and (3.20) changes the sign in (3.128). Finally, we find that on the subspace of the Killing spinors satisfying (3.126), ∂ ∂φ = − Γ 69 , ∂ ∂ψ = − 3 2 Γ 69 . (3.129) Together with (3.123), this gives us two additional algebraic equations, which as we will see simplifies the calculations significantly. We should also note that both projectors do not depend on the choice of the square root branch in (3.21) used to eliminate A 0 . For the RG flows, taking the limit `→∞ eliminates the first term in (3.125). The space of solutions includes then both Γ 12 -chiralities; the projector Π 3 is thus absent and we have a four-dimensional space of solutions corresponding toN = 2 supersymmetry. We have shown in [19] that the remaining two commuting projectors in this limit are Π ∞ 2 = 1 2 1 + (cosα Γ 5 − sinα Γ 4 ) Γ 69 (cosω Γ 10 + sinω Γ 11 ) , (3.130) and Π ∞ 0 = 1 2 1 + cosβ Γ 123 + sinβ (cosα Γ 4 + sinα Γ 5 ) , (3.131) 58 where the angles α, β and ω are some functions of r and χ. 15 In analogy with (3.127), the projector (3.130) can be associated with an extension of the complex structure of CP 2 to an almost complex structure with extra pair of complex frames. Finally, (3.131) is the dielectric deformation of the standard M2-brane projector at β = 0. For the Janus solutions, the projectors (3.130) and (3.131) must be deformed to account for the defect, which gives rise to additional terms in the supersymmetry variations, such as the first term in (3.125). Including such terms in (3.130) and (3.131) leads to the following Ansatz for the projectors at finite `: Π 0 = 1 2 1 +a 1 Γ 3 +a 2 Γ 4 +a 3 Γ 5 , (3.132) and Π 2 = 1 2 1 + (b 1 Γ 3 +b 2 Γ 4 +b 3 Γ 5 ) Γ 69 (cosω Γ 10 + sinω Γ 11 ) . (3.133) Those two operators form a pair of commuting projectors provided the vectors a≡ (a 1 ,a 2 ,a 3 ) andb≡ (b 1 ,b 2 ,b 3 ) are orthonormal. Such a pair of vectors can be parametrized by three angles, α, β and γ: a 1 = cosβ cosγ− sinα sinβ sinγ, a 2 = cosα sinβ, a 3 = sinα sinβ cosγ + cosβ sinγ, (3.134) and b 1 = − cosα sinγ, b 2 = − sinα, b 3 = cosα cosγ. (3.135) Together with ω those angles are some functions of r and χ and will be determined by solving the supersymmetry variations. For γ = 0, the projectors Π 0 and Π 2 reduce to Π ∞ 0 and Π ∞ 2 , respectively. We can thus view the angle γ as the Janus deformation parameter which goes to zero in the RG-flow limit. There is still certain redundancy in our description of the projectors (3.132) and (3.133). 15 See, (4.8) and (4.5) in [19]. 59 To see this, introduce a third vector, c, so that (a,b,c) are orthonormal and define x· Γ≡ (x 1 Γ 3 +x 2 Γ 4 +x 3 Γ 5 ). Observe that the product (b· Γ)(c· Γ)Γ 10 Γ 11 commutes with all the projectors, Π 0 ,..., Π 3 and so preserves the space of supersymmetries. One is therefore free to rotate (3.132) and (3.133) using the action of (b· Γ)(c· Γ)Γ 10 Γ 11 and this induces a simultaneous rotation ω→ ω +ϑ accompanied by a rotation of b and c by the angle ϑ. In the following we will use this freedom to simplify our calculations. 3.5.2 Supersymmetries for the Janus solutions We will now calculate all the projectors and the Killing spinor, , by solving explicitly the BPS equations (3.120). In principle, one should be able to determine all the projectors in (3.126), or equivalently solve for the angles α, β,γ and ω, directly from (3.124). The problem is that this effectively amounts to obtaining the individual projectors Π 0 ,..., Π 3 from a particular linear combination of products of these projectors. This, unsurprisingly, is not the best way to proceed. Instead, we will first solve algebraic equations that arise from judicious linear combinations of the variations (3.120) in which the flux terms either cancel completely or are simple. The first such equation arises from the “magical combination” of variations 2 Γ 1 δψ 1 + Γ 6 δψ 6 + Γ 7 δψ 7 + Γ 10 δψ 10 + Γ 11 δψ 11 = 0, (3.136) in which all flux terms cancel. After eliminating the derivatives with respect to the U(1) angles using (3.129) and modulo terms annihilated by Π 1 and Π 3 , it reads h A 1 Γ 3 +A 2 Γ 4 +A 3 Γ 5 + Γ 69 (A 4 Γ 10 +A 5 Γ 11 ) i = 0, (3.137) 60 where A 1 = 1 Σ 1/3 X 1/6 e −A ` , A 2 = A 0 Σ 1/3 X 1/6 , A 3 = m 7 X 1/3 (2 cos(2χ)− 1) sin(2χ) Σ 1/3 , A 4 = − m 7 X 1/3 (cos(2χ)− 2)) sin(2χ) Σ 1/3 , A 5 = m 7 X 1/3 (X(cos(2χ)− 2) + 3 Σ) 2 cos 2 χ Σ 1/3 . (3.138) Iterating (3.137) one finds a single consistency condition A 2 1 +A 2 2 +A 2 3 −A 2 4 −A 2 5 = 0, (3.139) which is satisfied by virtue of (3.21) and (3.116). This condition also means that, up to an invertible factor, (3.137) is in fact the projector (3.133) with b 1 = A 1 A , b 2 = A 2 A , b 3 = A 3 A , cosω = A 4 A , sinω = A 5 A , (3.140) where A≡ (A 2 1 +A 2 2 +A 2 3 ) 1/2 = (A 2 4 +A 2 5 ) 1/2 . (3.141) Using (3.135) and (3.140), we then read off cosα cosγ = − (2 cos(2χ)− 1)X 1/2 Ω 1/2 , cosα sinγ = − a sin(2χ) Ω 1/2 e −A ` , sinα = − a sin(2χ)A 0 Ω 1/2 , (3.142) and cosω = (cos(2χ)− 2)X 1/2 Ω 1/2 , sinω = sin(2χ)(3p 2 −X 2 + 3) 2X 1/2 Ω 1/2 , (3.143) where Ω = (1− 2 cos(2χ)) 2 X + 2 sin 2 (2χ)W 2 . (3.144) 61 Before proceeding we note that the rotation of the gamma matrices that define the projectors is equivalent to a rotation of the frames. In particular, the rotation byω is equivalent to starting with the frames: ˆ e 10 ≡ cosωe 10 + sinωe 11 , ˆ e 11 ≡ − sinωe 10 + cosωe 11 . (3.145) Using (3.143) we find a rather simple result for one of these frames: ˆ e 10 = m 7 X 1 6 Σ 1 3 Ω − 1 2 dφ + 3 2 (dψ + 1 2 sin 2 θσ 3 ) . (3.146) Note that the mixing ofφ andψ does not involve functions ofr and furthermore (3.129) implies that the supersymmetries only depend upon angles in precisely the combination (φ + 3 2 ψ). We will return to this observation later. Continuing with the supersymmetry analysis, since the projectors (3.132) and (3.133) com- mute, we cannot obtain any information from (3.137) about the dielectric polarization angle, β. For that we turn to another magical combination, Γ 1 δψ 1 + Γ 7 δψ 7 + Γ 8 δψ 8 = 0, (3.147) which has no derivatives of and no terms with components of the internal flux. After imposing the constant projections, it reads h B 1 +B 2 Γ 3 +B 3 Γ 4 +B 4 Γ 5 + Γ 69 (B 5 Γ 10 +B 6 Γ 11 ) i = 0, (3.148) where B 1 = m 7 p Σ 3/2 X 2/3 , B 2 = 1 2 Σ 1/3 X 1/6 e −A ` − pX 0 −p 0 X 2X , B 3 = 2XA 0 +X 0 4 Σ 3/2 X 7/6 , B 4 = −B 5 = − m 7 X 1/3 tanχ Σ 1/3 , B 6 = m 7 Σ 1/3 X 2/3 . (3.149) 62 Note that the presence of the B 1 -term in (3.148), with an analogous term absent in (3.137), prevents (3.148) from being a projector. Still, by iteration one finds a consistency condition B 2 1 −B 2 2 −B 2 3 −B 2 4 +B 2 5 +B 2 6 = 0, (3.150) which is indeed satisfied by virtue of (3.101) and (3.21). Usingtheprojectors(3.132)and(3.133)in(3.148), oneisleftwiththreeindependentproducts of Γ-matrices which yield the following equations: B 1 (sinα sinβ cosγ + cosβ sinγ) + (B 5 cosω +B 6 sinω) cosα cosγ + (B 5 sinω−B 6 cosω) (sinα cosβ cosγ− sinβ sinγ)−B 4 = 0, (3.151) B 2 (sinα sinβ cosγ + cosβ sinγ) +B 4 (sinα sinβ sinγ− cosβ cosγ) + (B 5 cosω +B 6 sinω) cosα cosβ + (B 5 sinω−B 6 cosω) sinα = 0, (3.152) B 3 (sinα sinβ cosγ + cosβ sinγ)−B 4 cosα sinβ− (B 5 sinω−B 6 cosω) cosα sinγ + (B 5 cosω +B 6 sinω) (sinα cosβ sinγ + sinβ cosγ) = 0. (3.153) However, only one of those equations is independent, which can be seen by solving one of them for tan(β/2) and then verifying that the other two are satisfied. Equivalently, one can solve the first two for cosβ and sinβ and then check that their squares indeed add up to one. Substituting the result into the third equations yields a consistency condition B 2 (cosβ cosγ− sinα sinβ sinγ) +B 3 cosα sinβ +B 4 (sinα sinβ cosγ + cosβ sinγ)−B 1 = 0. (3.154) This equation has a simple geometrical interpretation, namely that the (non-unit) vector d ≡ B 2 B 1 , B 3 B 1 , B 4 B 1 , (3.155) 63 satisfies a·d = − 1, (3.156) which is the consistency condition between the operator in (3.148) and the projector (3.132). Similarly as for the equations of motion in section 3.4, the solution for cosβ and sinβ above can be simplified using rationalized BPS equations. After some algebra, we find the following result: cosβ = cosγC 0 + sinγC 1 , sinβ = cosγS 0 + sinγS 1 , (3.157) where C 0 = − m −1 7 4X 3/2 Σ X 2 + 3 X 2 sin 2 χ + cos 2 χ +p 2 X 2 (cos(2χ)− 2) + 6 cos 2 χ + 3p 4 cos 2 χ A 0 W 2 , S 0 = p Ω 1/2 √ 2 Σ A 0 W 2 , (3.158) and C 1 = p sin(2χ) 8 ΣXW 2 −X 4 + (16 cos(2χ)− 5(cos(4χ) + 3)) csc 2 (2χ)X 2 + 6p 2 X 2 − 3 − 9p 4 − 9 , S 1 = − √ Ω csc(2χ) 4 ΣX 3/2 W 2 X 4 sin 2 (χ) +X 2 p 2 − 1 (cos(2χ)− 2) + 3 p 2 + 1 2 cos 2 (χ) . (3.159) This completes the calculation of all the angles in the projectors Π 0 and Π 2 . To determine explicitly the Killing spinors for unbroken supersymmetries, let us introduce rotations R ij (x) = cosx− sinx Γ ij , i,j > 1, (3.160) and define R(α,β,γ,ω) =R 35 (γ/2)R 45 (α/2)R 34 (β/2)R 10 11 (ω/2), (3.161) 64 which commute with the projectors Π 1 and Π 3 . It is straightforward to check that the projectors (3.132) and (3.133) are then simply Π 0 =R(α,β,γ,ω) Π (0) 0 R(α,β,γ,ω) −1 , Π 2 =R(α,β,γ,ω) Π (0) 2 R(α,β,γ,ω) −1 , (3.162) where Π (0) 3 = 1 2 (1 + Γ 3 ), Π (0) 4 = 1 2 (1 + Γ 569 10 ). (3.163) Thus any solution to (3.126) can be written as =R(α,β,γ,ω) ˜ , (3.164) where ˜ is in the kernel of the constant projectors (3.127), (3.128) and (3.163). From the supersymmetry variations along the radial direction, y, we find ∂˜ ∂y = 1 2` ˜ , (3.165) which is the correct radial dependence for the Killing spinor along AdS 3 . This leaves two variations along r and χ, which are solved as usual by setting ˜ = H 1/2 0 ε, (3.166) where H 0 = X 1/6 Σ 1/3 e A(r) , (3.167) is the warp factor of the “time” frame, e 1 = H 0 dt, and ε is a constant spinor along the inter- nal manifold and with the standard dependence along AdS 3 , which satisfies the same constant projections as ˜ . 65 3.5.3 The RG-flow limit The supersymmetry analysis simplifies significantly for the holographic flow solution. For this one simply imposes the projectors (3.127), (3.130) and (3.131) but does not impose a helicity projector like (3.132). We then find, taking the upper signs in (3.22) and (3.23): cosα = (2 cos(2χ)− 1)X 1/2 Ω 1/2 , sinα = − a sin(2χ)A 0 Ω 1/2 , (3.168) cosω = (cos(2χ)− 2)X 1/2 Ω 1/2 , sinω = sin(2χ)(3p 2 −X 2 + 3) 2X 1/2 Ω 1/2 , (3.169) and cosβ = − 1 2 √ 2WX 3/2 Σ X 2 + 3 X 2 sin 2 χ + cos 2 χ +p 2 X 2 (cos(2χ)− 2) + 6 cos 2 χ + 3p 4 cos 2 χ , sinβ = p Ω 1/2 √ 2W Σ . (3.170) The space-time components of the Maxwell fields also simplify and we obtain a seemingly stan- dard relation for holographic flows: h 0 = − 1 2 cosβ. (3.171) 3.6 IR asymptotics in eleven dimensions Having constructed the uplift in detail, we now examine the infra-red limits of the holographic RG flows described by (3.22) and (3.23) from the perspective of M theory. In an earlier paper [19] we focussed upon the special flow with ζ = π/3 since this led to a very interesting new result. Here we will complete the asymptotic analysis for all flows. First recall that ζ limits to a constant value as λ→ +∞ and so the various warp factors 66 behave as follows: X ∼ 1 2 (1 + cosζ)e 2λ , Ξ ∼ cosζe 2λ , Σ ∼ 1 2 e 2λ b Σ, b Σ ≡ (1− cosζ cos 2χ). (3.172) 3.6.1 cos 3ζ6=−1 For such a generic ζ one has dλ ∼ ∓ g 4 q (1 + cos 3ζ)e 3λ dr, e A ∼ R 2 e −λ , (3.173) for some constant, R. Thus the warp factor for the branes and the corresponding frames are finite and smooth for ζ6= 0,π: e i ∼ 1 √ 2 R 2 (1 + cosζ) 1 6 b Σ 1 3 dx i , i = 1, 2, 3 ; (3.174) Thus the metric parallel to the branes is simply: ds 2 3 = 3 X i=1 (e i ) 2 ∼ (1 + cosζ) 1 3 R 4 b Σ 2 3 (−dx 2 1 +dx 2 3 +dx 2 3 ) . (3.175) From (3.92) and (3.172) one has e 4 ∼ ∓ 2 √ 2 g (1 + cosζ) 1 6 b Σ 1 3 (1 + cos 3ζ) 1 2 e −2λ dλ, e 11 ∼ 2m 7 (1 + cosζ) 1 3 b Σ 2 3 e −2λ dφ + cos 2 χ (dψ + 1 2 sin 2 θσ 3 ) , (3.176) These are the only two frames to depend on λ in this limit. 67 The remaining frames limit to: e 5 ∼ m 7 b Σ 1 + cosζ 1 3 dχ ; e 6 ∼ m 7 b Σ 1 + cosζ − 1 6 cosχdθ ; (3.177) e 7 ∼ m 7 2 b Σ 1 + cosζ − 1 6 cosχ sinθσ 1 ; (3.178) e 8 ∼ m 7 2 b Σ 1 + cosζ − 1 6 cosχ sinθσ 2 ; (3.179) e 9 ∼ m 7 2 b Σ 1 + cosζ − 1 6 cosχ sinθ cosθσ 3 ; (3.180) e 10 ∼ m 7 b Σ 1 + cosζ − 2 3 sinχ cosχ (dψ + 1 2 sin 2 θσ 3 ) + 2 cosζ (1 + cosζ) dφ . (3.181) It is instructive to rewrite e 11 in terms of the one-form appearing in e 10 : e 11 ∼ 2m 7 b Σ 1 3 (1 + cosζ) 4 3 e −2λ dφ + (1 + cosζ) cos 2 χ b Σ dψ + 1 2 sin 2 θσ 3 + 2 cosζ (1 + cosζ) dφ , (3.182) Then one has ds 2 2 = (e 4 ) 2 + (e 11 ) 2 ∼ m 2 7 (1 + cosζ) 1 3 b Σ 2 3 (1 + cos 3ζ) dρ 2 +ρ 2 4 (1 + cos 3ζ) (1 + cosζ) 3 × × dφ + (1 + cosζ) cos 2 χ b Σ dψ + 1 2 sin 2 θσ 3 + 2 cosζ (1 + cosζ) dφ 2 , (3.183) where ρ≡e −2λ . The remaining part of the metric is ds 2 6 = 10 X j=5 (e j ) 2 ∼ m 2 7 b Σ 1 + cosζ 2 3 dχ 2 + (1 + cosζ) cos 2 χ b Σ ds 2 CP 2 + (1 + cosζ) 2 b Σ 2 sin 2 χ cos 2 χ dψ + 1 2 sin 2 θσ 3 + 2 cosζ (1 + cosζ) dφ , (3.184) 68 The full eleven-dimensional metric limits to the sum of (3.175), (3.183) and (3.184). Observe that (3.184) is conformally Kähler. That is, the metric d ds 6 2 = b Σ (1 + cosζ) dχ 2 + cos 2 χds 2 CP 2 + (1 + cosζ) b Σ sin 2 χ cos 2 χ dψ + 1 2 sin 2 θσ 3 + 2 cosζ (1 + cosζ) dφ , (3.185) has a Kähler form: b J ≡ − sinχ cosχdχ∧ dψ + 1 2 sin 2 θσ 3 + 2 cosζ (1 + cosζ) dφ + cos 2 χJ CP 2 = d 1 2 cos 2 χ dψ + 1 2 sin 2 θσ 3 + 2 cosζ (1 + cosζ) dφ , (3.186) where J CP 2 is the Kähler form onCP 2 . Here we are, of course, taking ζ to be constant at its asymptotic value. One can also easily verify that as χ→ π/2 this manifold is smooth, and is locally like the origin ofR 6 . The only singular parts of the metric occur at ρ = 0 and at χ = 0, where there are orbifold singularities in two different R 2 planes in (3.183) and (3.184) respectively. As we will discuss below, these loci represent the intersections of the various branes that are present in the infra-red limit. The non-zero components of the Maxwell field are given by: A (3) ∼ h 0 (r,χ)e 1 ∧e 2 ∧e 3 + 1 4 sinζ (e 6 ∧e 9 +e 7 ∧e 8 −e 5 ∧e 10 )∧ ˆ e 11 , (3.187) with h 0 ∼ sign(1− 2 cosζ) (cosζ− cos 2χ) b Σ , (3.188) ˆ e 11 ≡ 2m 7 (1 + cosζ) 1 3 b Σ 2 3 dφ + cos 2 χ (dψ + 1 2 sin 2 θσ 3 ) . (3.189) 69 ThusA (3) has regular coordinate components. One might be concerned that the Maxwell tensor has a singular source at ρ = 0 because the e 11 is vanishing. However, the frame components of the Maxwell tensor are, in fact, regular. The non-zero frame components in the compactified directions (including e 11 ) are: F 46911 ,F 47811 ∼ − 2m 7 sign(1− 2 cosζ) b Σ 4 3 (1 + cosζ) 1 3 (cos 2χ cosζ− sin 2 χ) tan 1 2 ζ, (3.190) F 451011 ∼ − 2m 7 sign(1− 2 cosζ) sinζ (1 + cosζ) 2 3 b Σ 1 3 , (3.191) F 56910 ,F 57810 ∼ − 2m 7 (1 + cosζ sin 2 χ) sinζ (1 + cosζ) 2 3 b Σ 4 3 , F 6789 ∼ 2m 7 sinζ (1 + cosζ) 2 3 b Σ 1 3 . (3.192) It is also useful to note that the electric part of the Maxwell field is extremely simple F electric (4) = dA e (3) , A e (3) = − 1 2 R 6 sign(1− 2 cosζ) cos 1 2 ζ cos 2χdx 1 ∧dx 2 ∧dx 3 . (3.193) 3.6.2 ζ =±π/3 Here we simply takeζ = +π/3 because the flow forζ =−π/3 simply involves reversing the sign of the internal components of the flux, A (3) . One now has rather different asymptotics: dλ ∼ ∓ g 2 √ 2 e λ dr, e A ∼ R 2 e −3λ , (3.194) ds 2 11 ∼ 2 − 4 3 3 1 3 b Σ 2 3 dρ 2 ρ 2 + ρ 2 R 2 (−dx 2 1 +dx 2 3 +dx 2 3 ) + 64 27 ρ 2 dφ + 3 cos 2 χ 2 b Σ (dψ + 1 2 sin 2 θσ 3 + 2 3 dφ) 2 + 4 3 m −2 7 dχ 2 + 3 2 b Σ cos 2 χds 2 CP 2 + 9 4 b Σ 2 sin 2 χ cos 2 χ dψ + 1 2 sin 2 θσ 3 + 2 3 dφ 2 , (3.195) 70 where, as before, ρ ≡ e −2λ , b Σ ≡ (1− 1 2 cos 2χ). (3.196) Note that the compact six-dimensional metric in (3.195) is simply the metric (3.184) specialized to ζ =π/3 and is therefore also conformally Kähler. Remarkably, for ζ =π/3 many of the components of F (4) vanish in the infra-red and we find that this limiting Maxwell field is simply given by F (4) = dA 0 (3) , A 0 (3) = √ 3m −3 7 4 b Σ cos 4 χ J CP 2 ∧ (dψ + 1 2 sin 2 θσ 3 + 2 3 dφ). (3.197) Note that the space-time components parametrized byh 0 vanish in this limit and thatF is purely magnetic and lives entirely on the conformally Kähler six-manifold. Thus, for ζ =π/3 there are only M5 branes in the infra-red: the M2 branes have dissolved completely. 3.6.3 The IR limit of the flows The first and rather remarkable surprise is that the warp factor, X 1 6 Σ 1 3 e A , in front of frames parallel to the M2-branes (3.92) is not singular in the infra-red for ζ6= 0,π. For ζ = 0,π, this warp factor is expected to be singular because such a flow has no internal fluxes and the warp factor is then simply a power of the harmonic function describing M2 brane sources that have spread on the Coulomb branch. However (3.174) shows that there is no singularity for generic ζ and for ζ =±π/3 equation (3.195) shows that this warp factor actually vanishes. Thus there are no strongly singular sources of M2 branes in the infra-red. The second surprise is that the internal six-dimensional manifold goes to a finite-sized con- formally Kähler, six-dimensional manifold and this manifold is smooth at χ =π/2. Indeed, the only singularities are conical and occur at ρ = 0 where the U(1) fiber defined by e 11 pinches off (see (3.183)) and atχ = 0 where the U(1) fiber defined bye 10 pinches off (see (3.184)). It is also evident from (3.188) and (3.190)–(3.192) that the core of this holographic flow is populated by finite, smooth electric (M2-brane) and magnetic (M5-brane) fluxes. Thus there is evidently brane 71 polarization and a geometric transition in which the M2 branes partially dissolve into smooth M5-brane fluxes leaving a finite sized “bubble” in the form of a six-dimensional Kähler manifold. Tounderstandthebranecontentintheinfra-redinmoredetailitisperhapseasiesttoexamine the projectors that define the supersymmetries. These are given by (3.127), (3.130) and (3.131). Define the rotated frames Γ b 4 ≡ cosα Γ 4 + sinα Γ 5 , Γ b 5 ≡ cosα Γ 5 − sinα Γ 4 , (3.198) Γ b 10 ≡ cosω Γ 10 + sinω Γ 11 , Γ b 11 ≡ cosω Γ 11 − sinω Γ 10 , (3.199) whereα =α(r,χ) andω =ω(r,χ) are functions that depend upon the flow. The details of these angles and how they flow are given in section 3.5.3 and may also be found in [19]. Given the other projectors and the fact that Γ 1...11 = 1, one can write (3.131) as Π 0 ≡ 1 2 (1 + cosβ Γ 123 + sinβ Γ Int ), (3.200) where Γ Int is any one of the following Γ 12369b 11 , Γ 12378b 11 , Γ 123b 5b 10b 11 , (3.201) This means that the flow represents M2 branes polarizing into three sets of M5 branes that have (3 + 1) common directions, those of the M2 branes and one compactified direction, defined by ˆ e 11 . This means that the directions transverse to the M5 branes are defined by ˆ e 4 , ˆ e 10 and four of the compact internal directions. Thus the brane wrapping is crucially determined by ˆ e 11 and hence by ω. For cos 3ζ6=−1 and λ→∞, one has: cosβ = cosζ− cosχ (1− cosζ cosχ) , α = ω = π 2 . (3.202) 72 Thus ˆ e 4 =e 5 , ˆ e 11 =−e 10 and so χ lies transverse to all the branes. Indeed, (3.202) shows that χ = 0 involves only anti-M2 brane sources and so the conical singularity at this point is not altogether surprising. The locusρ≡e −2λ = 0 also defines the location of the residual M2 branes and of some of the M5 branes and thus another conical singularity is not surprising. All the M5 branes have a common direction along e 10 , which is the Hopf fiber in the Kähler metric (3.185). One rather interesting flow involves having ζ→ π/2 at infinity. This does not mean that ζ = π/2 all along the flow; indeed (3.26) takes the value− 1 2 on such a flow and this implies that as λ→ 0 one must have ζ→ arccos(± 1 √ 5 ). What makes this flow interesting is that SU(4) symmetry is restored in the infra-red. In particular, the metric (3.185) becomes precisely that ofCP 3 As described in [19], the situation is very different for cos 3ζ =−1. Forζ =π/3 andλ→∞, one has 16 : cosβ = 0, ω = 0, (3.203) and cosα = 2 cos 2χ− 1 2− cos 2χ , sinα = − √ 3 sin 2χ 2− cos 2χ . (3.204) We now have ˆ e 11 =e 11 and so the M5 branes wrape 11 whilee 10 remains transverse to the branes. More significantly, the M2-brane flux now vanishes entirely and all that remains is a very simple non-singular magnetic (M5-brane) flux (3.197). The limiting metric (3.195) is almost like that of AdS 5 ×B 6 whereB 6 is the conformally Kähler metric. The five-dimensional manifold that we label as [ AdS 5 is AdS 5 in Poincaré form with one spatial direction compactified and fibered overB 6 . Holographically it suggests that the IR phase is almost a CFT except that one spatial direction has been “put in a periodic” box of some fixed scale and that some interactions have been turned on so that this direction becomes non-trivially fibered. Thus the IR phase is almost a CFT fixed point. Finally, we would like to note that all the fluxes and most, if not all, of the metric in the IR 16 The solution for ζ =−π/3 simply flips the signs of the internal fluxes and is completely equivalent. 73 limit are purely functions ofχ. This however, does not mean that these limits represent solutions to the equations of motion because ther (orλ) dependence is critical to giving finite terms that survive in the IR limit of the equations of motion. 3.7 Conclusion We have seen how apparently very singular “Flows to Hades” in gauged supergravity can encode some very interesting physical flows and Janus solutions when lifted to M-theory. We have found an interesting example of a supersymmetric flow withζ =π/3 which exhibits a remarkable infra- red limit in which only M5-brane fluxes survive and whose dual field theory is almost conformal in 3 + 1 dimensions. The infra-red geometry, [ AdS 5 ×B 6 , is smooth up to possible orbifold singularities at fixed points of the U(1) orbits. The other supersymmetric flows (ζ6= π/3) are also interesting but rather different from the one with ζ =π/3. For genericζ, there is still some M2 flux surviving in the infra-red limit, the compactification manifold,B 6 , is still conformally Kähler and the scale factors for this part of the metric and the metric along the M2 branes limits to a finite value. The radial coordinate and the remaining U(1) fiber conspire to make an orbifold ofR 2 fibered overB 6 . There is no hint of conformal behavior in the infra-red and the U(1) fiber scales to zero size while the M2-brane part of the metric retains a finite scale, which means that the infra-red limit is still intrinsically a (2 + 1)-dimensional field theory on the M2 branes. For holography, the class of flows considered here are extremely interesting. They start from the (2 + 1)-dimensional conformal field theory of M2 branes and “flow up in dimensions” to a field theory on the common (3 + 1)-dimensional intersections of M5 branes. Moreover, the theory on the M5 branes is almost conformal: the AdS 5 has one spatial direction compactified and fibered. The compactification is relatively straightforward but presumably the non-trivial fibration corresponds to adding background fields and twisting the theory on the M5 branes. We expect that this signals a new strongly-coupled phase of the holographic field theory on the M2 branes in which families of charged solitonic objects are becoming massless. It would 74 be extremely interesting to understand this in more detail for the M2-brane theory and for its orbifolds that lead to the ABJM theory [30]. 75 Chapter 4 Microstate geometries in five dimensions Inthischapterwereviewtheconstructionofthree-chargedmicrostategeometriesoffive-dimensional supersymmetric black hole. The material of this chapter is contained in [73]. 4.1 Brane configuration and supersymmetries We start with configuration of branes that preserves the same supersymmetries as the three- charge black hole in five dimensions. We take three sets of M2 branes that wrap three orthogonal two-tori of the T 6 . Preserving the same amount of supersymmetry one can add three sets of magnetic M5 branes that wrap the T 4 orthogonal to T 2 wrapped by corresponding M2 branes. The spatial directions of M5 branes wrap the closed curve y μ (σ). The branes are smeared over their transverse directions inside T 6 . The brane configuration is summarized in the table 4.1. The metric corresponding to this configuration is ds 2 11 =ds 2 5 + Z 2 Z 3 Z −2 1 1 3 (dx 2 5 +dx 2 6 ) + Z 1 Z 3 Z −2 2 1 3 (dx 2 7 +dx 2 8 ) + Z 1 Z 2 Z −2 3 1 3 (dx 2 9 +dx 2 10 ), (4.1) 76 Brane 0 1 2 3 4 5 6 7 8 9 10 M2 l ? ? ? ? l l ↔ ↔ ↔ ↔ M2 l ? ? ? ? ↔ ↔ l l ↔ ↔ M2 l ? ? ? ? ↔ ↔ ↔ ↔ l l M5 l y μ (σ) ↔ ↔ l l l l M5 l y μ (σ) l l ↔ ↔ l l M5 l y μ (σ) l l l l ↔ ↔ Table 4.1: Layout of the branes that give the supertubes and black rings in an M-theory duality frame. Vertical arrowsl, indicate the directions along which the branes are extended, and horizontal arrows, ↔, indicate the smearing directions. The functions,y μ (σ), indicate that the brane wraps a simple closed curve inR 4 that defines the black-ring or supertube profile. A star, ?, indicates that a brane is smeared along the supertube profile, and pointlike on the other three directions [73]. where the five-dimensional space-time metric has the form: ds 2 5 ≡ − (Z 1 Z 2 Z 3 ) − 2 3 (dt +k) 2 + (Z 1 Z 2 Z 3 ) 1 3 h μν dx μ dx ν , (4.2) where k is the one-form defined on the spatial section of the metric. We will find it convenient to define a four-dimensional spatial baseB with metric h μν , which is required to be asymptotic to Euclidean metric ofR 4 . The Killing spinor satisfies the standard projection conditions for M2 branes: 1 l + Γ 056 ) = 1 l + Γ 078 ) = 1 l + Γ 09 10 ) = 0. (4.3) Every projector reduces the amount of supersymmetries in half, so the total number of super- charges is four, i.e. 1 8 BPS. Since the product of all the gamma-matrices is the identity matrix, this implies 1 l − Γ 1234 ) = 0. (4.4) The Killing spinor on the four dimensional space must satisfy the covariant constraint ∇ μ = 0. (4.5) 77 The integrability condition for Killing spinor can be written as [∇ μ ,∇ ν ] = 1 4 R (4) μνcd Γ cd , (4.6) where R (4) μνcd is the Riemann tensor of four-dimensional metric h μν . It is satisfied automatically for spinors obeying (4.4) if the if the Riemann tensor is self-dual: R (4) abcd = 1 2 ε cd ef R (4) abef . (4.7) The holonomy of the general four-dimensional Euclidean metric is SU(2)×SU(2), and (4.7) implies that the holonomy lies in one of the SU(2) factors. The condition (4.4) means that all the components of the supersymmetry upon which the non-trivial holonomy would act vanish. In four dimensions SU(2) holonomy is equivalent to requiring that the metric be hyper-Kähler. There is a theorem that states that if the metric is Riemannian with signature +4, hyper- Kähler and asymptotic to R 4 , then it has to be globally the flat R 4 . However, this theorem has two important exceptions. First, one can change the asymptotic of the metric from R 4 to R 3 ×S 1 , which allow to interpret our solution in terms of asymptotically flat four-dimensional space-time instead of asymptotically flat five-dimensional space-time. Second, the four-metric h μν can be allowed to change its signature from +4 to−4 since this can be compensated by a sign change in the warp factors of (4.2) to produce a smooth five-dimensional space-time. Such four-dimensional metrics are called ambipolar. 4.2 BPS equations The three form potential is given by C (3) =A 1 ∧dx 5 ∧dx 6 + A 2 ∧dx 7 ∧dx 8 + A 3 ∧dx 9 ∧dx 10 , (4.8) 78 where A I ,I = 1, 2, 3 are Maxwell potentials in five-dimensional theory. It is convenient to introduce the Maxwell “dipole field strengths”, Θ I , obtained by separating the contributions of the electrostatic potentials Θ I = dB I , A I = −Z −1 I (dt +k) +B I , (4.9) where vector potential B I is on spatial baseB alone. The most general supersymmetric configuration is then obtained by solving the BPS equa- tions: Θ I = ? 4 Θ I , (4.10) ∇ 2 Z I = 1 2 C IJK ? 4 (Θ J ∧ Θ K ), (4.11) dk + ? 4 dk = Z I Θ I , (4.12) where? 4 is the Hodge dual taken with respect to the four-dimensional metrich μν , and structure constants are C IJK ≡| IJK |. If the equations of the system are solved in the right order then each of them is linear, and the non-linearity comes only from the sources. 4.3 Gibbons-Hawking space The supersymmetry allows the four-dimensional base-space metric to be hyper-Kähler. The par- ticular class of hyper-Kähler metrics with tri-holomorphic U(1) isometry is given by Gibbons- Hawking spaces. They provides examples of asymptotically locally Euclidean (ALE) and asymp- totically locally flat (ALF) spaces, which are asymptotic toR 4 /Z n andR 3 ×S 1 respectively. Gibbons-Hawking metrics can be written as the U(1) fibration over a flatR 3 base: h μν dx μ dx ν = V −1 dψ + ~ A·d~ y 2 + V (dx 2 +dy 2 +dz 2 ), (4.13) 79 where we write ~ y = (x,y,z). The function, V, is harmonic on the flatR 3 and the connection, A = ~ A·d~ y, is related to V via ~ ∇× ~ A = ~ ∇V . (4.14) The potential V is taken to be the sum of isolated sources: V =ε 0 + N X j=1 q j r j , (4.15) where r j ≡|~ y−~ y (j) |. The charges q j have to be integers in order for the metric to be single- valued, and space looks locally likeR 4 /Z |q j | near the points where r j = 0. For the metric to be Riemannian allq i have to be non-negative while for the ambipolar metricq i can be both positive and negative. If 0 6= 0 then the metric is asymptotic to flatR 3 ×S 1 , and one can do the standard Kaluza-Klein reduction to relate it to physics in four spacetime dimensions. If 0 = 0 then at large r the metric is asymptotic to flatR 4 /Z |q 0 | , where q 0 is the sum of all charges q j . Thus, to have a globally non-trivial metric, which asymptotes toR 4 at infinity, one has to allow q j to be negative integers such that the total charge q 0 = 1. 4.4 Magnetic fluxes on Gibbons-Hawking space The Gibbons-Hawking space withN centers has 1 2 N(N− 1) topologically non-trivial two-cycles. These cycles are formed by theU(1) fiber and the curveγ ij running between the GH centers~ y (i) and ~ y (j) . The cycles are topologically two-spheres up toZ k identification, and they intersect at common~ y (i) points, see Fig. 4.1. There is the basis of (N− 1) linearly independent cycles of the form Δ i(i+1) . It is also convenient to introduce a set of frames ˆ e 1 = V − 1 2 (dψ + A), ˆ e a+1 = V 1 2 dy a , a = 1, 2, 3. (4.16) 80 y (i) y (j) y (k) Δ ij Δ jk R 3 Figure4.1: Thisfiguredepictssomenon-trivialcyclesoftheGibbons-Hawkinggeometry. Thebehaviour of the U(1) fiber is shown along curves between the sources of the potential, V. Here the fibers sweep out a pair of intersecting homology spheres [73]. and two associated sets of two-forms: Ω (a) ± ≡ ˆ e 1 ∧ ˆ e a+1 ± 1 2 abc ˆ e b+1 ∧ ˆ e c+1 , a = 1, 2, 3. (4.17) The two-forms, Ω (a) − , are anti-self-dual, harmonic and they define the hyper-Kähler structure on the base. The forms, Ω (a) + , are self-dual and can be used to construct harmonic fluxes that are dual to the two-cycles. Consider the self-dual two-form: Θ ≡ 3 X a=1 ∂ a H V Ω (a) + . (4.18) Then Θ is closed if and only if H is harmonic inR 3 , i.e. ∇ 2 H = 0. To obtain the geometry without singularities H has to have its sources located at the same GH points as V: H = h 0 + N X j=1 h j r j . (4.19) There is also a gauge transformation, which means that the function H can be shifted by any amount of V without changing Θ: H → H + cV , (4.20) where c is the constant. If 0 = 0 then we must set h 0 = 0 to have a finite Θ at infinity. That amounts to (N− 1) independent regular harmonic two-forms, which are dual to (N− 1) independent homology two-cycles. 81 One can introduce the one-form magnetic potential B such that Θ =dB: B ≡ H V (dψ + A) + ~ ξ·d~ y, (4.21) where ~ ∇× ~ ξ = − ~ ∇H. (4.22) To find explicitly the vector potential it is convenient to introduce vectors~ v i : ~ ∇×~ v i = ~ ∇ 1 r i ! . (4.23) The vector fields are given by ~ A = N X j=1 q j ~ v j , ~ ξ = − N X j=1 h j ~ v j . (4.24) If one chooses the GH point to be~ y (i) = (0, 0,a), then in cylindrical coordinates the we will have ~ v i ·d~ y = (z−a) r i + c i dφ, (4.25) where c i is a constant. By choosing constants c i we can move the location of Dirac string along thez-axis. The circlesψ andφ shrink at~ y (i) , and the magnetic potential around~ y (i) is dominated by the term B ∼ h i q i dψ +q i (z−a) r i +c i dφ − h i (z−a) r i +c i dφ ∼ h i q i dψ. (4.26) The flux threading the cycle Δ ij can be obtained as follows: Π ij ≡ 1 4π Z Δ ij Θ = h j q j − h i q i ! . (4.27) On an ambipolar space the potential B and flux Θ are divergent at the surface V = 0, so the 82 previous integral is rather formal object. However, it is possible to show that after one imposes the regularity conditions, which comes form requiring the absence of closed time-like curves, the physical flux of Maxwell field, dA I , is regular and given by the same expression. 4.5 Solution of BPS equations The process of solving BPS equations (4.10)-(4.12) on Gibbons-Hawking space starts with defin- ing the three sets of self-dual harmonic two-forms Θ I [74]. As it was explained in the previous section that can be done by introducing three harmonic functions K I onR 3 . Then it is easy to check that the solution of the second-layer equation (4.11) is given by Z I = 1 2 C IJK K J K K V + L I , (4.28) where the L I are three more independent harmonic functions. The solution of the third-layer equation (4.12) can be written as k = μ (dψ +A) + ω, (4.29) with μ = 1 6 C IJK K I K J K K V 2 + 1 2 K I L I V + M, (4.30) where M is another harmonic function onR 3 . One-form ω satisfies the first order equation ~ ∇×~ ω = V ~ ∇M − M ~ ∇V + 1 2 (K I ~ ∇L I −L I ~ ∇K I ). (4.31) The integrability condition for this equation is satisfied automatically, since functions K I ,L I ,M and V are harmonic. 83 4.6 Closed time-like curves For the solution to be physically meaningful one must ensure the absence of the closed time-like curves (CTC). To do this we look at the space-like slices of the metric by keepingt constant. The absence of CTS’s along the T 6 directions immediately gives Z I Z J > 0. The four-dimensional slice of the metric reduces to: ds 2 4 = −W −4 μ(dψ +A) +ω 2 + W 2 V −1 dψ +A 2 +W 2 V dr 2 +r 2 dθ 2 +r 2 sin 2 θdφ 2 , (4.32) where we defined the warp-factor W≡ (Z 1 Z 2 Z 3 ) 1/6 . The metric might have singularity at the surfaceV = 0; however, the expansion of the metric shows that the potential singularity cancels out, and the metric is completely regular. Expansion of (4.32) leads to: ds 2 4 = Q W 4 V 2 dψ +A− μV 2 Q ω 2 +W 2 V r 2 sin 2 θdφ 2 − ω 2 Q +W 2 V (dr 2 +r 2 dθ 2 ), (4.33) where we have introduced the quantity: Q ≡ W 6 V − μ 2 V 2 = Z 1 Z 2 Z 3 V − μ 2 V 2 . (4.34) It turns out thatQ is a quartic invariant ofE 7(7) duality group ofN = 8 ungauged supergravity in four dimensions obtained by reducing M-theory on T 7 . To avoid CTC’s we have to impose the following conditions: Q ≥ 0, W 2 V ≥ 0, Z J Z K Z −2 I 1 3 = W 2 Z −1 I ≥ 0, I = 1, 2, 3. (4.35) 84 The last two conditions can be subsumed into: V Z I = 1 2 C IJK K J K K + L I V ≥ 0, I = 1, 2, 3. (4.36) 4.7 Bubbled solution The microstate geometry can be obtained by resolving the singularity of the brane configuration such as black ring trough the process known as the “geometric transition”. If one starts with the branes wrapping some cycle and increases the gravitational coupling the branes will back- react on the geometry and shrink the cycle to the zero size. For a black ring the M5 branes are wrapping the curve y μ (σ) inR 4 which is topologically S 1 . At the same time the dual “Gaussian cycle” which measures the flux carried by the branes becomes large and topologically non-trivial. The resulting geometry has a different topology, and no brane sources. The only information about the branes is now in the integral of the flux over the blown-up dual “Gaussian cycle”. It can be shown that if the solution before the transition has tri-holomorphic U(1) isometry then the resulting geometry must be the Gibbons-Hawking one. The singularity is resolved by the nucleation, or “pair creation”, of two equal and oppositely charged Gibbons-Hawking points. In the three-charge microstate geometry the charges of M2 branes are completely dissolved in M5-brane fluxes which are supported by blowing up the non-trivial two cycle on the Gibbons- Hawking space. The Chern-Simons interaction plays a key role in this phenomenon: as it is clearly seen from the second layer equation (4.11) the electric potentialZ I can be sourced by two other magnetic fluxes Θ J and Θ K . The resulting solution has therefore a non-trivial topology, no M2-brane sources and can be made completely regular (see Fig 4.2). FinallyletusshowhowtoconstructtheregularsolutionwiththeambipolarGibbons-Hawking base. To define regular fluxes Θ I we need to specify three sets of a harmonic functions K I : K I = k I 0 + N X j=1 k I j r j . (4.37) 85 Resolved Solution Naive Solution O O ring b a Figure 4.2: Geometric transition of black ring: The first diagram shows the geometry before the tran- sition. The second shows the resolved geometry, where a pair of GH charges has nucleated at positions a and b. The harmonic functions L I and M must be chosen such that the warp-factors, Z I , and function μ are regular at GH pionts, i.e. at r j → 0. That implies that we must take: L I = ` I 0 + N X j=1 ` I j r j , M = m 0 + N X j=1 m j r j , (4.38) with ` I j = − 1 2 C IJK k J j k K j q j , j = 1,...,N ; (4.39) m j = 1 12 C IJK k I j k J j k K j q 2 j = 1 2 k 1 j k 2 j k 3 j q 2 j , j = 1,...,N. (4.40) To solve the last equation for ~ ω it is convenient to introduce vectors ω ij that satisfy: ~ ∇×~ ω ij = 1 r i ~ ∇ 1 r j − 1 r j ~ ∇ 1 r i + 1 r ij ~ ∇ 1 r i − ~ ∇ 1 r j ! , (4.41) where r ij ≡ |~ y (i) − ~ y (j) | (4.42) is the distance between thei th andj th center in the Gibbons-Hawking metric. In the coordinate system chosen such that ~ y (i) = (0, 0,a) and~ y (j) = (0, 0,b) the vectors ω ij have the form ω ij = − (x 2 +y 2 + (z−a +r i )(z−b−r j )) (a−b)r i r j dφ, (4.43) 86 Then the general solution of (4.31) can be written as: ~ ω = N X i,j a ij ~ ω ij + N X i b i ~ v i , (4.44) forsomeconstantsa ij ,b i . Itispossibletocheckusingtheexplicitexpression(4.43)thatfunctions ω ij have no Dirac-Misner strings, and all string singularities of ~ ω can come only from functions ~ v i . Thus, in general there are N possible string singularities that can be arranged to run in any direction from each of the GH points ~ y (j) . Finally, if we want our solution to be asymptotic to five-dimensional Minkowski space, we have to take 0 = 0 and k I 0 = 0. Moreover, μ must vanish at infinity, and this fixes m 0 . We also normalize the size of the T 2 by taking Z I → 1 at r→∞. That results in the constraints: ε 0 = 0, k I 0 = 0, l I 0 = 1, m 0 =− 1 2 q −1 0 N X j=1 3 X I=1 k I j . (4.45) 4.8 Bubble equation To have a globally well-defined causal structure of spacetime one has to make sure that it does not contain closed timelike curves (CTS’s). A CTC is a forward-directed path that is always timelike which intersects itself at a point in its “past”. At the end of Section 4.6 we formulated the conditions for the absence of CTC’s that have to be satisfied globally: Q ≥ 0, V Z I = 1 2 C IJK K J K K + L I V ≥ 0, I = 1, 2, 3. (4.46) It is not known how to verify these conditions in general, but it is possible to formulate the set of equations that in many instances ensure that (4.46) are satisfied. They are called bubble equations, and follow from the analysis of the solution at Gibbons-Hawking points, i.e. atr j → 0. In the previous section we fixed the harmonic functionsL I andM such that functionsZ I ,μ, and therefore W, limit finite values at r j → 0 while V −1 vanishes. By looking at (4.32) we see 87 that that in order to avoid CTC in ψ-direction we need to impose N conditions: μ(~ y =~ y (j) ) = 0, j = 1,...,N. (4.47) Another potential CTC may come from the φ-circle if the form ω has a finite dφ component at θ = 0 or θ = π. This is equivalent to the presence of a Dirac-Misner string singularity, so we must ensure that the final solution do not have them. It turns out that both set of conditions are actually the same. The string singularity in ~ ω comes from the term μ ~ ∇V, and therefore removing them leads to conditions (4.47). Performing the expansion of μ using (4.30), (4.37), (4.38) and (4.40) around each Gibbons- Hawking point we find that (4.47) becomes the Bubble Equations: N X j=1 j6=i Π (1) ij Π (2) ij Π (3) ij q i q j r ij = −2 m 0 q i + 1 2 3 X I=1 k I i . (4.48) As we saw in the previous section the string singularities of ~ ω come only form the sum of ~ v i terms, see (4.44). If one collects all the terms that contributes to this sum and set the coefficients to zero,b i = 0, one ends up with the same set of equations (4.48). The sum of the left-hand side of bubble equations vanishes automatically, while the sum of the right-hand sides vanishes due to the condition for m 0 in (4.45). Thus there are only (N− 1) independent bubble equations. Although bubble equations provide a necessary conditions to avoid CTC’s around Gibbons- Hawking points, they do not guarantee the absence of CTC’s globally. Moreover, there are examples when bubbled equations are satisfied, but the solution still has CTC’s. However, in many physically important examples the bubbled equations are enough to guarantee the global absence of CTC’s. In some cases it is possible to numerically establish the absence of CTC’s. In general the question when the bubbled solution is free of CTC’s is still open. 88 Chapter 5 Supersymmetry and wrapped branes in microstate geometries 5.1 Motivation New class of states that can carry the significant portion of entropy of the three charged black hole has been discovered recently by Martinec and Niehoff [18]. In M-theory these states come from the M2 branes wrapping non-trivial cycles in the deep, scaling microstate geometries where they become very light. Authors of [18] pointed out that within the compactified directions, the wrapped branes behave as point particles in the magnetic fluxes that thread the compact directions, and so there is a vast number of distinct W-branes coming from the degeneracy of lowest Landau level. It was also suggested that such “W-branes” could condense and give rise to new phases with an exponentially growing number of BPS states. Up until recently, W-branes have been treated as probes in the background of microstate geometries. However, iftherearesufficientlymanywrappedbranesthentherewillbeasignificant back-reaction and this will require treatment within supergravity. It is therefore possible that supergravity may, in fact, see some large-scale, coherent aspects of W-branes. The simplest approach to investigating such wrapped branes in supergravity is to start with 89 the T 6 compactification of M-theory and consider an M2 brane wrapping an S 2 in a five- dimensional microstate geometry. Such a brane is point-like in the T 6 but one can simplify the problem by smearing over the entire torus and reducing the problem to five-dimensional supergravity. The smearing also avoids the problem of how to handle the electric field lines on the compactification manifold since it forces all the electric field lines into the space-time. In five dimensions, the four-form field strength sourced by such a wrapped brane is dual to a scalar field and the relevant five-dimensional field theory isN =2 supergravity coupled to both vector multiplets and hypermultiplets. Thus far, the study of microstate geometries from the five-dimensional perspective has largely focused onN = 2 supergravity coupled only to vector multiplets. The addition of hypermultiplets introduces a whole new level of complexity but is required if one is to study wrapped brane states in five dimensions. The new family of backreacting M2 branes wrapping S 2 cycle in AdS 3 ×S 2 were recently studied by Raeymaekers and Van den Bleeken in connection with the black hole deconstruction proposal [75]. This new family was shown to preserve the same amount of supersymmetries as microstates and thus has to represent the backreacted version of W-brane. However, it was not clear how those supersymmetries would be modified in an asymptotically flat background and how the supersymmetries might depend upon the orientation of one bubble relative to another. In this chapter we show that wrapping M2 branes on a space-time 2-cycles will generically breakallthesupersymmetries. Wearguethattheonlysupersymmetricconfigurationsare 1 16 -BPS withtwosupersymmetries, andtheyinvolveM2braneswrappingasinglecycleorseveralco-linear cycles with the same orientation. Therefore, the solutions found in [75] should not be considered as microstates of 1 8 -BPS black holes, since they break some or all of its supersymmetries. 90 5.2 The Lagrangian and BPS equations Weworkwithinfive-dimensional,N =2supergravitycoupledtobothvectorandhypermultiplets. The bosonic action may be taken to be: S = Z √ −gd 5 x R− Q IJ ∂ μ X I ∂ μ X J − 1 2 h uv D μ q u D μ q v − 1 2 Q IJ F I μν F Jμν − 1 24 C IJK F I μν F J ρσ A K λ ¯ μνρσλ . (5.1) Our goal is to write the action in a manner that is a simple extension ofN = 2 supergravity coupled to vector multiplets that is typically used in the discussion of microstate geometries. Our space-time metric is “mostly plus” and we will only have two vector multiplets and hence three vector fields. Thus I,J = 1, 2, 3, and we normalize the A I so that C 123 = 1. The scalars satisfy the constraint X 1 X 2 X 3 = 1 and metric for the kinetic terms is: Q IJ = 1 2 diag (X 1 ) −2 , (X 2 ) −2 , (X 3 ) −2 . (5.2) As usual, it is convenient to introduce three scalar fields, Z I , and take Z ≡ (Z 1 Z 2 Z 3 ) 1/3 , X J ≡ Z Z J , X J ≡ 1 3 Z J Z . (5.3) The scalars, q u , are those of the hypermultiplets. One can easily relate our conventions most simply to those of [76]. Define ˆ A I ≡ − √ 3A I , ˆ C IJK ≡ 1 6 C IJK , h I ≡ X I , h I ≡ X I , a IJ ≡ 2 3 Q IJ . (5.4) then the hatted quantities are those of [76] and we have set κ = 1 √ 2 . The conventions of [77] are very similar, except they use a “mostly minus” metric and thus one must send g μν →−g μν and modify gamma matrices appropriately. The BPS equations come from setting all the supersymmetry variations of the fermions to 91 zero: ∇ μ i + i 8 X I F I νρ γ μνρ − 4g μν γ ρ i − ∂ μ q v ω v ij j = 0, (5.5) iγ μ ∂ μ X I + 1 2 δ I J − X I X J F I ρσ γ ρσ i = 0, (5.6) iγ μ (D μ q v )f jA v j = 0. (5.7) The symplectic indices are raised and lowered using v i = ij v j , v i = v j ji , (5.8) and our gamma matrices satisfy n γ a ,γ b o = 2η ab , γ abcde = i abcde , 01234 ≡ +1, (γ 0 ) † =−γ 0 , (γ A ) † = (γ A ) † , A = 1, 2, 3, 4. (5.9) 5.3 The standard bubbled geometries 5.3.1 Two centers and AdS 3 ×S 2 If one can separate two of the GH centers from the rest and if they are close enough together so that one can ignore the constants, ε 0 and ` 0 , then the resulting space-time may be reduced to AdS 3 ×S 2 [78–80]. The GH potential is simply: V = q + r + − q − r − , (5.10) where q ± ≥ 0 and r ± ≡ q ρ 2 + (z∓a) 2 . (5.11) 92 Gauge transformations allow us to shift K I →K I +c I V, which means we can shift the poles in the K I and assume, without loss of generality, that K I = k I 1 r + + 1 r − . (5.12) By uplifting to six dimensions one can shift V by one of the K I ’s and such a spectral flow can be used to set V = q 1 r + − 1 r − , (5.13) For simplicity, we will take: V = q 1 r + − 1 r − , K I = K = k 1 r + + 1 r − , (5.14) L I = L = − k 2 q 1 r + − 1 r − , M = − 2k 3 aq 2 + 1 2 k 3 q 2 1 r + + 1 r − , (5.15) where the forms of the L I and M are determined by regularity. One then finds Z I = Z = V −1 K 2 +L = − 4k 2 q 1 (r + −r − ) , (5.16) μ = V −2 K 3 + 3 2 V −1 KL +M = 4k 3 q 2 (r + +r − ) (r + −r − ) 2 − 2k 3 aq 2 . (5.17) The one forms are given by: A = q (z−a) r + − (z +a) r − dφ, ~ ξ·d~ y = −k (z−a) r + + (z +a) r − dφ, ω = − 2k 3 aq ρ 2 + (z−a +r + )(z +a−r − ) r + r − dφ. (5.18) This metric (4.2) is equivalent to theAdS 3 ×S 2 space-time and can be written in the global form by performing the following coordinate transformation: z =a cosh 2ξ cosθ, ρ =a sinh 2ξ sinθ, ξ≥ 0, 0≤θ≤π, (5.19) 93 and shifting and rescaling variables: τ ≡ aq 8k 3 t, ϕ 1 ≡ 1 2q ψ− aq 8k 3 t, ϕ 2 ≡ φ− 1 2q ψ + aq 4k 3 t. (5.20) The metric (4.2) then takes the simple form: ds 2 5 ≡ R 2 1 h − cosh 2 ξdτ 2 +dξ 2 + sinh 2 ξdϕ 2 1 i + R 2 2 h dθ 2 + sin 2 θdϕ 2 2 i , (5.21) where R 1 = 2R 2 = 4k. (5.22) The Maxwell field also dramatically simplifies and reduces to: A = −2k cosθdϕ 2 , F = dA = 2k sinθdθ∧dϕ 2 , (5.23) and so, as one would expect, F is proportional to the volume form on the S 2 . 5.3.2 Other frames and coordinates There is another standard form of the metric onAdS 3 that will prove useful: The Bergman form, which describes the metric as a non-trivial time fibration over a non-compact Kähler base. The AdS 3 factor of (5.21) can be written as ds 2 3 ≡ R 2 1 4 h − (d ˆ t + 2 sinh 2 1 2 ζd ˆ ψ) 2 + dζ 2 + sinh 2 ζd ˆ ψ 2 i , (5.24) This comes from a very simple change of variable in (5.21): ξ = 1 2 ζ, τ = 1 2 ˆ t, ϕ 1 = ( ˆ ψ− 1 2 ˆ t). (5.25) 94 In particular, this and (5.20) implies ˆ ψ = 1 2q ψ. (5.26) It will be convenient to introduce three sets of frames for each of the three forms of the metric: e 0 = Z −1 (dt +μ(dψ +A) +ω), e 1 = Z 1/2 V −1/2 (dψ +A), e 2 = Z 1/2 V 1/2 dρ, e 3 = Z 1/2 V 1/2 ρdφ, e 4 = Z 1/2 V 1/2 dz ; (5.27) ˜ e 0 = 4k coshξdτ, ˜ e 1 = 4kdξ, ˜ e 2 = 4k sinhξdϕ 1 , ˜ e 3 = 2kdθ, ˜ e 4 = −2k sinθdϕ 2 ; (5.28) ˆ e 0 = 2k (d ˆ t + 2 sinh 2 1 2 ζd ˆ ψ), ˆ e 1 = 2kdζ, ˆ e 2 = 2k sinhζd ˆ ψ, ˆ e 3 = 2kdθ, ˆ e 4 = −2k sinθdϕ 2 . (5.29) The negative sign in ˜ e 4 and ˆ e 4 might seem unusual but it is there to ensure that the Lorentz transformation from the frames (6.6) to (5.28) or (5.29) has determinant equal to +1. It is a trivial exercise to verify ˆ e 0 = coshξ ˜ e 0 + sinhξ ˜ e 2 , ˆ e 2 = sinhξ ˜ e 0 + coshξ ˜ e 2 . (5.30) The Lorentz transform between the e a and the ˜ e a is given by: e 0 = 1 cosθ h coshξ ˜ e 0 + sinhξ ˜ e 2 + sinθ ˜ e 4 i , e 1 = (coshξ sinη tanθ + sinhξ cosη) ˜ e 0 + (sinhξ sinη tanθ + coshξ cosη) ˜ e 2 + sinη cosθ ˜ e 4 , e 3 = (sinhξ sinη− coshξ cosη tanθ) ˜ e 0 + (coshξ sinη− sinhξ cosη tanθ) ˜ e 2 − cosη cosθ ˜ e 4 , e 2 = sinη ˜ e 1 + cosη ˜ e 3 , e 4 = cosη ˜ e 1 − sinη ˜ e 3 . (5.31) where cosη ≡ sinh 2ξ cosθ q cosh 2 2ξ− cos 2 θ , sinη ≡ cosh 2ξ sinθ q cosh 2 2ξ− cos 2 θ . (5.32) 95 5.3.3 Killing spinors We continue with all the hypermultiplet scalars set to zero. Since our background obeys the “floatingbraneAnsatz"[81]theBPSequation(5.6)istriviallysatisfiedasaresultofacancellation between the connection terms and the Maxwell field strengths. This leaves the equation ∇ μ i + i 8 X I F I νρ γ μνρ − 4g μν γ ρ i = 0, (5.33) which determines how all the supersymmetries depend upon the coordinates. Indeed, using the ˜ e a frames (5.28) with (5.9) to write products of three gamma matrices in terms of products of two gamma matrices, we find the following differential equations: ∂ τ j = −∂ ϕ 1 j = 1 2 sinhξγ 01 j − 1 2 coshξγ 12 j , ∂ ξ j = 1 2 γ 02 j , ∂ θ j = − i 2 γ 4 j , ∂ ϕ 2 j = − 1 2 cosθγ 34 j − i 2 sinθγ 3 j . (5.34) One can trivially solve for the dependence onξ andθ and the rest can be solved direct by taking derivatives and commuting gamma matrices through the first part of the solution. We find j = e 1 2 ξγ 02 e − i 2 θγ 4 e 1 2 (ϕ 1 −τ)γ 12 e − 1 2 ϕ 2 γ 34 j 0 . (5.35) where j 0 is a constant spinor. Note that there are eight solutions: four components and two choices for j. These solutions contain both the Poincaré and superconformal supersymmetries. Ifoneusesthe ˆ e a frames(5.29)thenthelocalLorentzrotation(5.30)undoestheξ-dependence and gives j = e − i 2 θγ 4 e 1 2 (ϕ 1 −τ)γ 12 e − 1 2 ϕ 2 γ 34 j 0 . (5.36) Based on (5.31), define the “gamma matrix:” Γ 0 = 1 cosθ h coshξγ 0 + sinhξγ 2 + sinθγ 4 i . (5.37) 96 Observe that if we take the γ a in this expression to be that gamma matrices in the ˜ e a frames in (5.28), then (5.31) implies that Γ 0 represents theγ 0 matrix of the GH frames, (6.6). The natural Poincaré projection condition in a generic GH space is given by taking Γ 0 j =±i j for one choice of sign. Acting with Γ 0 on the spinor in (5.35) gives: Γ 0 j = e 1 2 ξγ 02 e − i 2 θγ 4 e 1 2 (ϕ 1 −τ)γ 12 e − 1 2 ϕ 2 γ 34 h γ 0 + tanθ (iγ 0 +1 l) i j 0 . (5.38) This implies that Γ 0 j = i j ⇔ γ 0 j 0 = i j 0 , (5.39) but that the solution space does not respect the projection with the opposite sign. The projection condition (5.39) therefore identifies the Poincaré supersymmetries associated with the general GH space. Note that this is normally recast using (5.9) so as to emphasize the hyper-Kähler property of the base: Γ 1234 j = j ⇔ γ 1234 j 0 = j 0 . (5.40) Alternatively, based on (5.30) and (5.31) one can take the γ a to be those of the Bergman frames, (5.29), and define the “gamma matrix:” b Γ 0 = 1 cosθ h γ 0 + sinθγ 4 i . (5.41) This is representative of the γ 0 matrix of the GH frames, (6.6), in the Bergman frames and, acting on (5.36), it leads to the same result as in (5.39) and (5.40). 5.4 The supersymmetries with hypermultiplet scalars 5.4.1 The hypermultiplet solutions The background considered in [75] is the half-hypermultiplet parametrized by a complex scalar, τ. The simplest way to satisfy the BPS equations for this is to takeτ =τ(z) to be a holomorphic 97 function of the coordinate, z = tanh ζ 2 e i ˆ φ on the Bergman base in (5.24). Indeed, the simplest non-trivial solution has: τ = −iq ∗ ln(z) + iV ∞ = q ∗ ˆ ψ + i(V ∞ −q ∗ ln(tanh ζ 2 )), (5.42) where q ∗ and V ∞ are constants. This locates the wrapped M2 branes at z = 0 with a source proportional to q ∗ . In [75] the solution is written in terms of coordinates (x,ψ) where: logz = x +i ˆ ψ ⇒ ζ =− log tanh − x 2 . (5.43) The new class of solutions obtained in [75] have a metric with frames E ˆ t = l 2 (d ˆ t− (1 + Φ 0 (x))d ˆ ψ), E ˆ x = l 2 √ τ 2 e −Φ(x) dx, E ˆ ψ = l 2 √ τ 2 e −Φ(x) d ˆ ψ, E ˆ θ = l 2 dθ, E ˆ φ = − l 2 sinθ (dφ−d ˆ ψ +d ˆ t ) = − l 2 sinθdϕ 2 . (5.44) where Φ(x) satisfies a non-linear, ordinary differential equation. Observe that we have made some changes of notation and convention compared to [75]. First, we have relabelledt andψ in [75] by ˆ t and ˆ ψ. This makes the notation in (5.44) consistent with our notation here and avoids the confusion between Im(log(z)) in (5.43) and the GH fiber coordinate, ψ. These coordinates are, of course, related by (5.26). We have also reversed the sign of ˆ t relative to t in [75] and flipped the orientation of the frame E ˆ φ . This brings (5.44) into line with the orientations of (6.6)–(5.29). It should be remembered that [75] uses conventions that make the two-forms in the BPS equations of bubbled geometries have the opposite dualities to the standard ones on the GH base. Our modifications restore the canonical forms of these duality conditions. If one sets q ∗ = 0 and thus removes the wrapped M2 branes then one has [75]: Φ(x) = ˜ Φ(x) + 1 2 logV ∞ , ˜ Φ(x) = log(sinh(−x)), τ 2 =V ∞ (5.45) 98 and one finds that the frames (5.44) become precisely the Bergman frames in (5.29) withl = 4k. 5.4.2 The supersymmetries In [75] it was shown that non-trivial half-hypermultiplet background imposes one additional projection condition on the supersymmetries if the AdS 3 ×S 2 background without the wrapped M2 branes. This condition is: 1 lδ i j − iγ ˆ x ˆ ψ σ 3 i j j = 0, (5.46) where σ 3 is the usual 2× 2 Pauli spin matrix acting on theN =2 indices of the spinor and γ ˆ x ˆ ψ refers to the product of gamma matrices in the frames (5.44). We first note that this is a projector in the Bergman basis and so must be applied to the Killing spinor (5.36). In particular, γ ˆ x ˆ ψ =γ 12 commutes with all the exponentials in (5.36) and thus implies: 1 lδ i j − iγ ˆ x ˆ ψ σ 3 i j j 0 = 0, (5.47) Next we observe that (5.9) implies that iγ ˆ x ˆ ψ = −γ ˆ t ˆ θ ˆ φ , (5.48) and so (5.46) is precisely the projector of a brane wrapping the S 2 . Using (5.30) and (5.31) one can also easily map this projector into standard GH form. To do this we note that E ˆ x ∧E ˆ ψ = ˆ e 1 ∧ ˆ e 2 = ˜ e 1 ∧ (sinhξ ˜ e 0 + coshξ ˜ e 2 ) = (sinηe 2 + cosηe 4 )∧ (cosηe 1 + sinηe 3 ) = − cos 2 ηe 1 ∧e 4 + sin 2 ηe 2 ∧e 3 − sinη cosη (e 1 ∧e 2 +e 3 ∧e 4 ), (5.49) 99 This means that in transforming from the Bergman basis to the GH basis, we have γ ˆ x ˆ ψ → − cos 2 ηγ 14 + sin 2 ηγ 23 − sinη cosη (γ 12 +γ 34 ). (5.50) However, because of the self-duality of the GH base and the projection condition (5.40) in the GH frames, we have γ ab j GH = − 1 2 abcd γ cd j GH , (5.51) and so (5.50) becomes iγ ˆ x ˆ ψ → iγ 23 = −γ 014 , (5.52) where we have used (5.9) in the last identity. Now recall that the homology cycles in a GH metric are defined by the ψ-circle fibered along any curve between poles of V. Moreover, the minimum area cycle involves the shortest such curve. Thus, in the GH form of the metric with (5.13), the two cycle is defined by the ψ-circle over the interval along the z-axis between−a and a. From (6.6), the area form of this cycle is e 1 ∧e 4 . Thus (5.52) corresponds to the projector for the M2 brane wrapping this cycle. In a general bubbled solution, each wrapped M2 brane will give rise to a supersymmetry projector that depends on the orientation of the brane. More precisely, the supersymmetry projector will depend upon the orientation of the straight line joining the two GH points in the baseR 3 parametrized by~ y in (4.2). Theγ 4 in (5.52) will be then replaced by a linear combination ofγ a ,a = 2, 3, 4. Any two such projectors are compatible (have a common null space) if and only if all the GH points are co-linear and the wrapped branes have the same orientation. Indeed, co-linear wrapped branes with opposite orientations source the Maxwell field with opposite signs and so lead to opposite signs in (5.47). A pair of such opposed projectors manifestly have no common null space. This has several important consequences for the supersymmetry. First, all the supersymmetry will be broken if the branes wrap cycles that are not co-linear. If the wrapped cycles are all co-linear then supersymmetry will still be broken if the branes wrap in different orientations, 100 determined by the relative signs of the Maxwell fields they source. This means that solutions with wrapped M2 brane but no net wrapped M2-brane charge necessarily break all the supersymmetries. Finally, if all the wrapped branes lie on co-linear cycles and have the same orientation then the projectors of these branes are all the same and the combined effect is that they reduce the supersymmetry by another factor of a half. As regards the total number of supersymmetries, the AdS 3 ×S 2 starts with eight real su- persymmetries once the symplectic Majorana condition is imposed on (5.35) or (5.36). If one simply wraps the S 2 , one preserves the conformal invariance and hence the superconformal su- persymmetries but one must impose the projector (5.47), and, as shown in [75], this leaves four supersymmetries. If one breaks the conformal invariance by either restoring the asymptotically flat region or by adding more bubbles then one must impose another supersymmetry projector, (5.39) or (5.40), which is compatible with the projector (5.47). This reduces the solution to two supersymmetries, and renders it a 1 16 -BPS background. 5.5 Conclusions We have shown that wrapped branes break some, or all, of the supersymmetries in a microstate geometryandthatthisisgovernedbythesupersymmetryprojectorsassociatedwiththewrapped brane. In particular, two of the four supersymmetries found in [75] are artefacts of the supercon- formal symmetry and will be lost as soon as the configuration is embedded in an asymptotically- flat space-time. In particular, if one takes a general 1 8 -BPS bubbled geometry and wraps any single bubble with M2 branes, the result is a 1 16 -BPS solution. If one wraps more than one bubble then all the supersymmetry will be broken unless all the wrapped bubbles are co-linear and are wrapped in the same orientation and only then will it be 1 16 -BPS. This means that such wrapped branes should not be identified with microstates of 1 8 -BPS black holes but should be viewed as (partial) supersymmetry-breaking excitations of such geometries. Martinec and Niehoff [18] point out that the fact that W-branes are becoming light in the 101 scaling limit means that there will be a new phase of stringy physics emerging in the deep scaling regime of microstate geometries. They argue that the W-branes will form condensates, and new operators will develop vevs and define order parameters in that new phase. Thus, one should think of W-branes as one-particle excitations on a new massless branch of physics that is opening up in deep scaling solutions and not as microstates. 102 Chapter 6 Non-trivial compactifications with dynamical moduli 6.1 Motivation One can try to put the microstate program in a broader supergravity compactification scheme. As we have argued above it is crucial to have the non-trivial topology in the spacetime to obtain a smooth horizonless solutions, while the topology of theT 6 remains fixed and plays an auxiliary role in this construction. On the other hand in the phenomenological applications of string theory to building models of particle physics one usually starts with ten-dimensional string theory compactified on the six-dimensional Calabi-Yau manifold. The particle spectrum and their interactions are then encoded in the non-trivial topology of the Calabi-Yau space, whereas the spacetime is kept the standard Minkowski one. At the same time it would be interesting to have the spacetime and compact space on the same footing, when both of them are topologically non-trivial. In the context of the microstate program it would be natural to allow for blowing up and down cycles not only in four spatial dimensions but in all ten such that the manifold remains regular. TheT 6 is obviously non easily adapted to this purpose since one cannot blow down the cycle without collapsing its volume. Hence, one would like to replace six-torus with the more 103 general manifold with a free blow-up parameter that does not make the whole compactification manifold singular. Another reason why it is interesting to consider the non-trivial topology of the compactifi- cation space comes from the attempt to construct the backreacted W-branes. In the previous chapter we have described one possible approach to this problem consisting of smearing the M2 brane wrapped on the spacial S 2 over the six-torus to avoid the problem with electric field lines. That gave us theN = 2 supergravity coupled to both vector and hypermultiplets. As we have shown the resulting solution in general breaks all the supersymmetries and therefore does not represent the microstate of the black hole. Thus, one would like consider an alternative possibility to keep the M2-brane probe particle as a localized source in the T 6 . If we increase the gravitational coupling the M2 brane will start backreacting on the geometry and shrinking the S 2 -cycle to zero size. As a result of the geometric transition the non-trivial homology cycle should blow up inside the compact manifold and the point-like source should be nucleated into the pair of equal and oppositely charged points. Such a pair creation will obviously avoid the problem with the field lines on compact manifold. To test this intuitive picture of the geometric transition in the calculable way one needs to have a good model of the compact manifold which is general enough to support non-trivial homology cycles. The simplest choice would be to replace T 4 inside T 6 by some four-manifold. The supersymmetry equations and the equations of motion tell us that this manifold has to be Kähler with the Ricci-flat metric, which means that the manifold has to be the Calabi-Yau two- fold. It is well known that the only non-trivial example of four-dimensional Calabi-Yau manifold is the K3 surface. The explicit metric on K3 is not known; however, as a local approximation one can take the multi-centered Gibbons-Hawking metric. Although the Gibbons-Hawking space is not compact, it can be considered as a local model of K3, and the approximation becomes better for large radius of K3. In this approximation one can explicitly manipulate with the moduli of K3 which control the size and shape of the cycles. In this chapter we consider an exact solutions of BPS equations for compactification of M- 104 theory onM GH ×T 2 , whereM GH is a Gibbons-Hawking ALE space. We focus on backgrounds with three electric charges in five dimensions that have the same four supersymmetries as the three-charge black hole. The analysis is quite non-trivial because the compactifying manifold must be fibered over the space-time using compensator fields that play the role of gauge con- nections on the moduli space ofM GH . We also analyze the equations of motions both at linear and non-linear order. We present the “topological solution,” for which the brane distribution on compactification manifold is always intrinsically dipolar (with no net charge) outside δ-function sources in the space-time. Such dipolar charge distributions lead to fields that fall off extremely rapidlywithdistanceandaveragetozeroonscalesthataremuchlargerthanthecompactification scale. Notation and conventions: Thischapterwillinvolvethereductionofeleven-dimensionalsupergravityonasix-dimensional, Riemannian internal manifold,Y, to a five-dimensional space-time,X. In particular,Y will ei- ther beT 6 orM GH ×T 2 . These spaces themselves will also involve spatial slices and fibrations, which will require further refinement of indices. In particular, since we are dealing with BPS solutions,X will come with a preferred time coordinate, t, and will find it convenient to write X as time fibration over a spatial base,B. To keep some order on this, we state many of our index conventions here. First, the eleven- dimensional space-time and frame indices will be M,N,P... and A,B,C..., respectively. We will typically use the indices μ,ν,... and i,j,... for tangent indices onB andY, and these indices will range over 1,..., 4 and 1,..., 6, respectively. The coordinates onX andY will be (t,x μ ) and y i . The frame indices onB will be α,β,... taking values 1,..., 4 (with 0 reserved for the time-like frame). We will also typically use a,b,···∈{1, 2, 3, 4} to be frame indices on M GH . However, in Section 6.3, we will consider compactifications on a generic manifoldX×Y, where the detailed structure is not important. In this section, and only in this section, μ,ν,... and α,β,... will be tangent and frame indices on all ofX and take values 0, 1,..., 4. Similarly, in Section 6.3, and only in this section, we will take a,b,···∈{1, 2,..., 6} to be frame indices 105 on a generic Calabi-Yau 3-fold,Y. 6.2 Geometric preliminaries to reducing onM GH ×T 2 In theT 6 =T 2 ×T 2 ×T 2 compactification, parametrized by coordinatesy 1 ,...,y 6 , we are going to replace T 2 ×T 2 factor by a more general manifold, e M. We note that on the T 2 ×T 2 defined by (y 1 ,y 2 ,y 3 ,y 4 ), (4.3) implies 1 l + Γ 5678 ε = 0. (6.1) The supersymmetry condition (6.1) means that the metric e ds 2 4 on e M must be hyper-Kähler with an anti-self-dual Riemann tensor: e R (4) abcd = − 1 2 ε cd ef e R (4) abef . (6.2) Thus the metric on the internal manifold, e M×T 2 , will still be Ricci-flat and Kähler and so falls into the broader class of Calabi-Yau compactifications. We can therefore employ all the technology that has been developed for such compactifications. The obvious choice for e M is K3 but, since the metrics on such manifolds are not explicitly known, we will take e M to be a Gibbons-Hawking (ALE) space,M GH , as a local model of aK3. SinceM GH is non-compact, it does not strictly represent a compactification, but we will abuse the terminology and still refer to it as a “compactification.” Despite the non-compactness ofM GH , we can still operate at the level of the equations of motionandusetheBPSequationsbecausethesearealllocalanddonotinvolvenon-normalizable integrals over space. Indeed, at this level, one can equally replaceT 2 ×T 2 byR 2 ×R 2 . When we consider effective actions, we will introduce a cut-off at large distances compared to the scales of the compact homology cycles of the GH manifold. We will therefore use the multi-centered, Riemannian GH metric with anti-self-dual Riemann 106 tensor: e ds 2 4 = V −1 (dψ +A) 2 + V (d~ z·d~ z), (6.3) where A = ~ A·d~ z and ~ ∇× ~ A = − ~ ∇V . (6.4) We take the potential, V, to be positive definite 1 and define “component functions,” K I2 , via: V = N X I=1 K I , K I ≡ q I r I , r I ≡ |~ z−~ z I |, (6.5) where q I ∈Z + . We also introduce the standard frames on the GH space: ˜ e 1 = V −1/2 (dψ +A), ˜ e ˆ a+1 = V 1/2 dz ˆ a , (6.6) where we have introduced indices ˆ a, ˆ b,···∈{1, 2, 3}. With this choice of frames, and the choice (6.4), the spin connection and the curvature are anti-self-dual, as required. The GH metric comes with three harmonic, self-dual Kähler forms: J (ˆ a) = Ω (ˆ a) + ≡ ˜ e 1 ∧ ˜ e ˆ a+1 + 1 2 ˆ a ˆ bˆ c ˜ e ˆ b+1 ∧ ˜ e ˆ c+1 . (6.7) In a Calabi-Yau compactification of M-theory, the vector multiplets are associated with the harmonic (1, 1)-forms of the compactification manifold. Our goal is to get to five-dimensional, N = 2 supergravity coupled to purely vector multiplets and so we have to choose a particular complex structure and then use it to identify the (1, 1)-forms. We choose the complex frames to be E 1 = ˜ e 1 + i ˜ e 4 , E 2 = ˜ e 2 + i ˜ e 3 , (6.8) 1 It would be extremely interesting to see if one can generalize this construction to ambi-polar GH metrics. 2 These functions should not be confused with the harmonic functions K I from chapters 4 and 5 which were defining the magnetic fluxes on GH base. 107 which means that the Kähler form is given by J = J (3) = i 2 (E 1 ∧E 1 +E 2 ∧E 2 ) = ˜ e 1 ∧ ˜ e 4 + ˜ e 2 ∧ ˜ e 3 = (dψ +A)∧dz 3 + V dz 1 ∧dz 2 . (6.9) The complex combinations,J (1) +iJ (2) andJ (1) −iJ (2) , are then harmonic (2, 0) and (0, 2) forms, respectively. In addition to complex structures, it is also useful to define three anti-self-dual two-forms: Ω (ˆ a) − = ˜ e 1 ∧ ˜ e ˆ a+1 − 1 2 ˆ a ˆ bˆ c ˜ e ˆ b+1 ∧ ˜ e ˆ c+1 . (6.10) The harmonic forms on Gibbons-Hawking space with N centers are given by: ω I = ∂ ˆ a K I V ! Ω (ˆ a) − =d " A I − K I V (dψ +A) # , (6.11) where ~ ∇× ~ A I = − ~ ∇K I . (6.12) Note that the forms in (6.11) are anti-self dual, normalizable and dual to the homology cycles of the GH manifold. One should also note that compared to the conversions of chapter 4, where GH manifold was the base manifold of the space-time, the self-duality and anti-self-duality of the forms and curvatures are interchanged. Also observe that, because of the first equation in (6.5), the ω I satisfy the constraint: N X I=1 ω I = 0. (6.13) There are thus only (N− 1) linearly independent such forms. SinceJ is self-dual and the Ω (ˆ a) − are anti-self-dual, it follows from the structure ofSO(4) that the matrix J a b commutes with the matrices Ω (ˆ a) − a b , where a,b,... are frame indices on the GH space. Hence: J a c Ω (ˆ a) − c b = Ω (ˆ a) − a c J c b ⇔ J a c J b d Ω (ˆ a) − cd = Ω (ˆ a) − ab . (6.14) 108 The second identity proves that all the harmonic forms given by (6.11) are, in fact, (1, 1)-forms with respect to J. The complete set of (1, 1)-forms onM GH is therefore spanned by linear combinations of{J,ω I } subject to the constraint (6.13). There are thusN such harmonic forms, and therefore the reduction onM GH will produce N massless vector fields, corresponding to N =2 supergravity coupled to (N− 1) vector multiplets. The analog of the T 6 projection conditions (4.3) can now be written onM GH ×T 2 as 3 : 1 l + 1 4 J ab Γ 0a+4b+4 ε = 1 l + Γ 5678 )ε = 1 l + Γ 09 10 )ε = 0. (6.15) The metric, (6.3), has 3(N− 1) moduli. These are parametrized by the ~ z I in (6.5) with the overall translation of the center of mass of the points~ z I being trivial. In principle, in a reduction onM GH , all of these moduli will correspond to massless scalars in the space-time. However, we are seeking a truncation with (N− 1) vector multiplets, and so we need to specialize to (N− 1) of these moduli. As we will discuss in Section 6.4, the (N− 1) moduli that we seek are precisely those that preserve the complex structure defined in (6.9), and they simply correspond to moving the GH points,~ z I , in the z 3 -direction. Finally, we recall that the simplest, non-trivial GH space hasN = 2 and is simply the Eguchi- Hansonspace. Therearethustwo (1, 1)forms,J andω. ReplacingT 2 ×T 2 bytheEguchi-Hanson space means that we replace torus 2-forms,dy 1 ∧dy 2 +dy 3 ∧dy 4 anddy 1 ∧dy 2 −dy 3 ∧dy 4 , byJ and ω respectively. Like their torus counterparts, J and ω also have trivial intersection: J∧ω = 0. Thus compactification on the product of an Eguchi-Hanson space with a T 2 should lead to the standard “STU” model, and be indistinguishable from theT 6 compactification, in the low-energy, five-dimensional limit. We now investigate how reduction on Calabi-Yau and GH manifolds works in more detail. 3 It turns that the second projector in (6.15) is redundant as it is the square of the first one. We include the second projector to simplify the exposition. 109 6.3 Generalities about Calabi-Yau compactification Our purpose here is to determine how the eleven-dimensional metric encodes the degrees of free- dom of five-dimensional,N =2 supergravity coupled to (N− 1) vector multiplets. In particular, the eleven-dimensional metric must encode the scalars of the vector multiplets as moduli of inter- nal, “compactification” metric. We start by considering the problem in a little more generality and take the internal manifold,Y, to be Calabi-Yau, with coordinates y i , Ricci-flat metric, e g ij , and moduli, u I . We also takeX to be a generic space-time with coordinates, x μ . Note that in this section, and only in this section, μ,ν,... and α,β,... will be tangent and frame indices on all ofX and take values 0, 1,..., 4 and that a,b,···∈{1, 2,..., 6} will be frame indices of the entire manifold,Y. Also note that, in order to encode the vector multiplet scalars, the metric onY will be allowed to depend onX but only through the moduli: u I =u I (x μ ). 6.3.1 The form of the eleven-dimensional metric The naive choice for the eleven-dimensional metric is ds 2 11 = g μν (x)dx μ dx ν +e g ij (y,u(x))dy i dy j , (6.16) While (4.1) has this form, this is too simple an Ansatz for more general internal metrics. As noted in [82], promoting the moduli to scalar fields on the space-time means that the metric must generically be fibered over the the space-time base,X. We therefore take the eleven-dimensional metric to have the form: ds 2 11 = g μν (x)dx μ dx ν +e g ij (y,u(x))(dy i −B i μ (y,u(x))dx μ )(dy j −B j ν (y,u(x))dx ν ). (6.17) The metric Ansatz in [82] omitted the quadratic terms in B i μ that appear in (6.17). This was sufficient for the linearized action obtained in that [82]. Our choice of (6.17) is motivated by the forms of fibrations that occur in consistent Kaluza-Klein Ansätze and, as we will see, our choice 110 is essential to satisfying the eleven-dimensional BPS equations. The fact that we are only going to allow the internal metric to depend on x μ through the moduli also passes into the fibration in that the vector fields must have the form: B i μ (y,u(x)) = B i I (y,u(x))∂ μ u I (x). (6.18) Indeed the B i I will be set equal to compensators onY, which are vector fields defined so to preserveregularityoftheexplicitmetriccomponentsunderderivativeswithrespecttothemoduli. 6.3.2 Compensators and Lichnerowicz modes Consider the family of smooth, Ricci-flat metrics on the internal manifold,Y: e ds 2 = ˜ g ij (y,u)dy i dy j , (6.19) which depends on some moduli,u I . By definition, changing theu I moves (6.19) through a family of smooth metrics, however this does not mean that differentiating the coordinate dependent quantity, ˜ g ij (y,u), with respect to some u I will produce a smooth result. What is guaranteed is that an infinitesimal shift in the u I , combined with an infinitesimal coordinate transformation, will translate (6.19) horizontally across the space of gauge orbits of smooth metrics, moving from one smooth, Ricci-flat metric to an infinitesimally neighboring smooth, Ricci-flat metric. Thus we need make the combined transformation: u I → u I + δu I , y i → y i +B i I (y,u)δu I , (6.20) for some appropriately-chosen compensating vector fields, B i I (y,u). Furthermore we define an associated covariant derivative D I = ∂ ∂u I +L B I , (6.21) 111 whereL B I is the Lie derivative along the compensating vector field B i I . It is these fields that we will use in the metric Ansatz (6.17). The fields B i I (y,u) are chosen so as to ensure that δ I g ij ≡ D I g ij = ∂ I g ij +L B I g ij = ∂ I g ij + (∇ I B jI +∇ j B iI ), (6.22) is a smooth, symmetric tensor field. This, however, does not fully specify B i I . The canonical way to fix these vector fields is to require that the variations, δ I g ij , are transverse and traceless: ∇ i δ I g ij = 0, g ij δ I g ij = 0. (6.23) Since the family of metrics is Ricci-flat, the variations, δ I g ij , are also zero-modes of the Lich- nerowicz operator, which, for transverse, traceless modes, can be written: ∇ k ∇ k δ I g ij − 2R kij` δ I g k` = 0. (6.24) On a four-dimensional Kähler manifold, one can relate the harmonic (1, 1) forms, ω, to Lichnerowicz zero-modes obtained by variations of the Kähler moduli by: δg ij = − 1 2 J i k ω kj + J j k ω ki . (6.25) For hyper-Kähler manifolds one may use any of the three complex structures, which gives rise to three different moduli for each harmonic form. For GH manifolds, each of theω I in (6.11) corresponds, in this way, to the three-component position vectors, ~ z I , in (6.5). Choosing the complex structure J (3) , as in (6.9), means that the metric variations of the form (6.25) using J = J (3) are Kähler deformations. The variations (6.25) that are then obtained from J (1) ,J (2) produce deformations of the complex structure defined by J (3) . As we will see in Section 6.4.1, the choice ofJ =J (3) , means that the third component,z 3 I of~ z I become the Kähler moduli while 112 the other components, z 1,2 I become complex structure moduli. 6.3.3 Frames and spin connections We now return to our eleven-dimensional metric, (6.17), and introduce frames and compute spin connections. It is useful to start by introducing orthonormal frames for the individual metrics, g μν and e g ij , in (6.17): ˆ e α = ˆ e α μ (x)dx μ , ˜ e a = ˜ e a i (y,u)dy i . (6.26) We define ˆ ω and ˜ ω to be the spin connections in these frames with the moduli,u I , considered as free, constant parameters (not depending on coordinates): dˆ e α = −ˆ ω α β ∧ ˆ e β , d˜ e a = −˜ ω a b ∧ ˆ e b . (6.27) We also define the individual components of these connections via: ˆ ω α β = ˆ ω γ α β ˆ e γ , ˜ ω a b = ˜ ω c a b ˜ e c . (6.28) We will also find it convenient to introduce the restricted exterior derivatives: d x ≡ dx μ ∧ ∂ ∂x μ d y ≡ dy i ∧ ∂ ∂y i . (6.29) We then take the frames of the fibered metric (6.17) to be: e α ≡ e α μ (x)dx μ = ˆ e α μ (x)dx μ , e a = ˜ e a −B a μ dx μ = ˜ e a i (y,u(x))dy i −B a μ dx μ . (6.30) 113 To write the spin connection explicitly, we define the tensors: F I a b ≡ ˜ e i b ∂˜ e a i ∂u I , F μ a b ≡ F I a b ∂ μ u I M I a b ≡ ˜ e i b D I ˜ e a i ≡ ˜ e i b ∂˜ e a i ∂u I + e ∇ b B a I , M α a b ≡ ˆ e μ α M I a b ∂ μ u I S αab ≡ 1 2 M αab +M αba , A αab ≡ 1 2 M αab −M αba , Y i IJ ≡ 1 2 ∂B i J ∂u I − ∂B i I ∂u J +B j I ∂ j B i J −B j J ∂ j B i I ! , Y a αβ ≡ ˆ e μ α ˆ e ν β ˜ e a i Y i IJ ∂ μ u I ∂ ν u J . (6.31) Onecanalsoeasilyverifythefollowingidentity, whichisusefulincomputingthespinconnections: Y a αβ ≡ ˆ e μ [α ˆ e ν β] ∂ μ B a ν +B b μ e ∇ b B a ν +B b μ F ν a b , (6.32) where [... ] represents skew-symmetrization of weight one. A straightforward calculation leads to the following components of the spin connection: ω α β = ω γ α β e γ +Y a α β e a , ω a b = ˜ ω c a b e c −A α a b e α , (6.33) ω a α = S α a b e b −Y a αβ e β . The symmetric part,S I (ab) , ofM I (ab) is, of course, the covariant derivative of the metric that leads to the Lichnerowicz mode: 2S I (ab) ˜ e a i ˜ e b j = ∂e g ij ∂u I + ( e ∇ I B jI + e ∇ j B iI ) = D I e g ij = δ I e g ij . (6.34) Theanti-symmetrictensors,A Iab , representlocalLorentzrotationsoftheframesthatareinduced by changes in the moduli. The tensor Y i IJ =−Y i JI may be thought of as the field strength of the gauge fields B i I considered as tangent vectors on the moduli space. In this perspective, the vector index,i, tangent toY is an internal index and the last two terms in the definition ofY i IJ in (6.31) make up the Lie derivative,L B I B J , and so may be thought of as structure constants in 114 the algebra of the B J . The reason for the fibration structure in the metric Ansatz, (6.17), has begun to emerge from (6.31)–(6.33). Without the compensators, the spin connections would merely involve partial derivatives, with respect to the u I , of the frames onY. Such partial derivatives are generically singular. Forregularity, derivativeswithrespecttomodulimustbepairedwiththecorresponding compensator, as in (6.21). As is evident from the computation above, the compensators in the fibration structure achieves this pairing and results in a regular spin connection. We will also see something similar with the flux Ansatz in Section 6.4.3. 6.4 Reduction onM GH ×T 2 Before we introduce our eleven-dimensional Ansatz and analyze the BPS equations for the com- pactification, we need a few more geometric details of the internal GH manifold. In particular, we need the compensators, B i I . Some of these results were obtained in [83] and we re-derive them and a make a minor correction. We also revert to our previous convention in whichμ,ν... and α,β,... are coordinate and frame indices onB (taking values 1,..., 4), x μ are coordinates onB, and a,b,... are frame indices onM GH , also taking values 1,..., 4. 6.4.1 Lichnerowicz modes, connections and compensators onM GH One can use (6.9) and (6.11) in (6.25) to construct the Lichnerowicz modes corresponding to Kähler deformations of the GH metric (6.3). One easily obtains: δ I e ds 4 ≡ 2S Iab ˜ e a ˜ e b = ∂ 3 K I V ! h (˜ e 1 ) 2 + (˜ e 4 ) 2 − (˜ e 2 ) 2 − (˜ e 3 ) 2 i + 2∂ 1 K I V ! h ˜ e 2 ˜ e 4 − ˜ e 1 ˜ e 3 i + 2∂ 2 K I V ! h ˜ e 1 ˜ e 2 + ˜ e 3 ˜ e 4 i , (6.35) 115 where we have used this result to read off the tensors, S Iab . More explicitly, one has: ω Iab = 2J ac S I c b . (6.36) To obtain these modes by differentiating the metric with respect to the moduli,~ z I , one must use the third components, z 3 I , combined with suitable compensators, B i I . It is relatively easy to infer the form of the compensators by noting that the potential, V, defined in (6.5) develops a stronger singularity if one simply differentiates by z 3 I . One can compensate for this by acting with an infinitesimal diffeomorphism that simultaneously moves the coordinate z 3 at~ z =~ z I but does not displace ~ z at ~ z J for J6= I. Such an infinitesimal diffeomorphism is generated by the vector field: B I = K I V ∂ 3 . (6.37) Indeed, these compensator vector fields were given in [83] and they suffice if all the GH points lie along the z 3 -axis. However, when the GH points lie in general positions, one must allow the fiber coordinate to undergo and infinitesimal gauge transformation: ψ→ψ +f I (~ z) and thus the complete compensator has the form B I = f I (~ z)∂ ψ + K I V ∂ 3 . (6.38) To fix the f I (~ z) we simply computeD I e ds 2 4 , whereD I is defined in (6.21): D I e ds 2 4 = ∂ 3 K I V ! h (˜ e 1 ) 2 + (˜ e 4 ) 2 − (˜ e 2 ) 2 − (˜ e 3 ) 2 i + 2∂ 1 K I V ! h ˜ e 2 ˜ e 4 − ˜ e 1 ˜ e 3 i + 2∂ 2 K I V ! h ˜ e 1 ˜ e 2 + ˜ e 3 ˜ e 4 i + 2V −1/2 d y f I + K I V A 3 −A I 3 ! ˜ e 1 . (6.39) Comparing this with (6.35) we conclude that one has to choose f I =A I 3 − K I V A 3 , where the A I 116 are defined in (6.12), and the complete compensators are therefore: B I = A I 3 − K I V A 3 ! ∂ ψ + K I V ∂ 3 . (6.40) If all GH centers lie on thez 3 -axis then the third components,A 3 ,A I 3 , of the vector fields vanish identically and we recover the expression for compensators found in [83]. One can also act withD I on the Kähler form, J, and find the expected relation: D I J = ω I . (6.41) We note, in passing, that since the Kähler deformations of the metric and complex structure are generated by z 3 I -derivatives, and not z 1,2 I -derivatives, this establishes that the z 3 I are indeed the Kähler moduli and that the z 1,2 I must be complex structure moduli. Thus the z 3 I represent scalars in the vector multiplets while the z 1,2 I are scalars in hypermultiplets. We will allow the former to be dynamical while the latter will remain fixed. The fact that we have obtained the compensators for generic locations of the GH points, rather than merely GH points along the z 3 -axis, means that we can describe the dynamics in a background with generic vevs of the hypermultiplet scalars. Finally, the tensor, F Iab , defined in (6.31), has the following non-zero components: F I 11 = F I 44 = −F I 22 = −F I 33 = 1 2 ∂ 3 K I V ! , F I 12 = F I 43 = ∂ 2 K I V ! , F I 42 = −F I 13 = ∂ 1 K I V ! . (6.42) Symmetrizing gives S I (ab) and the result matches with the expression (6.35). Skew symmetriza- tion leads to the components of the anti-symmetric tensor A Iab : 1 2 A Iab ˜ e a ∧ ˜ e b = 1 2 ∂ 2 K I V ! h ˜ e 1 ∧ ˜ e 2 − ˜ e 3 ∧ ˜ e 3 i − 1 2 ∂ 1 K I V ! h ˜ e 1 ∧ ˜ e 3 + ˜ e 2 ∧ ˜ e 4 i , (6.43) 117 which is anti-self-dual. It is not difficult to check that the tensor Y i IJ defined by (6.31) vanishes identically for the compensating fields (6.40). The fields B I are, in this sense, pure gauge. 6.4.2 The metric Ansatz We will take the full eleven-dimensional metric Ansatz to be a warped fibration ofM GH ×T 2 over a five-dimensional space-time,X. We will require the metric onX to be precisely those of the dimensionally reduced five-dimensional supergravity solutions. On the internal manifold, there is a warp factor that determines the relative volume of theM GH and the T 2 . This scalar is part of a vector multiplet in five dimensions and so it must be allowed to depend non-trivially upon the coordinates,x μ , ofX. The other vector multiplet scalars are the (N−1) Kähler moduli ofM GH and so these can also depend onx μ . The overall volume ofM GH ×T 2 represents a scalar in the five-dimensional universal hypermultiplet and so this degree of freedom will be frozen. For simplicity, we will also assume that the background is purely electrically charged and has no angular momentum. Our metric Ansatz is therefore: ds 2 11 = −Z −2 dt 2 +Zds 2 4 + Z Z 0 e ds 2 4 + Z 0 Z 2 (dy 2 5 +dy 2 6 ), ds 2 4 = g μν (x)dx μ dx ν , (6.44) e ds 2 4 = e g ij (y,u(x))(dy i −B i μ (y,u(x))dx μ )(dy j −B j ν (y,u(x))dx ν ), The functionsZ andZ 0 depend only on the coordinatesx μ onX, and the BPS condition means that all the fields must, of course, be time-independent. The factors ofZ in the space-time metric are motivated directly by (4.2) and the fact that the eleven-dimensional solution must reduce to that of five-dimensional supergravity. There could, in principle, have been another warp-factor, W (x μ ,y i ), multiplying the overall, five-dimensional space-time metric. However, we found that such a function was rendered trivial by the BPS conditions and so we have not included it here. We have also considered other 118 generalizations of this Ansatz but the ultimate justification of our choice (6.44) is that we will show that it is sufficient to solve the BPS equations. We introduce frames e 0 =Z −1 dt, e α =Z 1/2 ˆ e α μ dx μ , e a+4 = Z Z 0 1 2 ˜ e a i (dy i −B i μ dx μ ), e 9,10 = Z 0 Z dy 5,6 , (6.45) for which the the spin connection is: ω 0 α = −Z −1/2 (d logZ) α e 0 , ω α β = Z −1/2 ˆ ω γ α β e γ +Z −1/2 (d logZ 1/2 ) β e α −Z −1/2 (d logZ 1/2 ) α e β + Z −1/2 Z −1/2 0 Y a [αβ] e a , ω a+4 b+4 = Z 0 Z ˜ ω c a b e c+4 −Z −1/2 A α [ab] e α , (6.46) ω a+4 α = Z −1/2 S αab e b+4 +Z −1/2 d log(Z/Z 0 ) 1/2 α −Z −1/2 Z −1/2 0 Y a αβ e β , ω 9,10 α = Z −1/2 (d log(Z 0 /Z)) α e 9,10 , where ˆ ω and ˜ ω are defined in (6.27) and (6.28). 6.4.3 The flux Ansatz Since we are assuming that the background is purely electrically charged, the usual Calabi-Yau Ansatz suggests that we should take C (3) = −Φ 0 dt∧J + X I Φ I dt∧ω I , (6.47) for some potential functions Φ 0 , Φ I . This is too simplistic. First we need to use the compensators to ensure thatF (4) =dC (3) is suitably smooth and so, as in the metric Ansatz, this means that we must fiber the Kähler form. We therefore introduce 119 the frame components of J via: J ≡ 1 2 J ab ˜ e a ∧ ˜ e b , (6.48) and define the form J 0 onX×Y by: J 0 ≡ 1 2 J ab (˜ e a −B a α ˆ e α )∧ (˜ e b −B b β ˆ e β ) =J + 1 2 J ab B a α B b β ˆ e α ∧ ˆ e β −J ab B b β ˜ e a ∧ ˆ e β , (6.49) where B a α ≡ ˜ e a i ˆ e μ α B i I ∂ μ u I . One can also fiber the ω I in a similar manner. The total exterior derivative of J 0 can be computed using (6.33) dJ 0 = (d x +d y )J 0 =J ab A α a c ˜ e c ∧ ˜ e b ∧ ˆ e α −J ab S α a c ˜ e b ∧ ˜ e c ∧ ˆ e α +J ab ˜ ω d a c ˜ e b ∧ ˜ e c ∧ ˜ e d . (6.50) Now recall that the spin-connection and A α a c are anti-self-dual (see (6.43)) and that the com- mutator of a self-dual matrix with an anti-self-dual one is zero. It follows that the first and the third terms in (6.50) vanish. Indeed, the vanishing of the last term simply represents the closure of the Kähler form. We therefore arrive at the simple result: dJ 0 =J ac S α c b ˜ e a ∧ ˜ e b ∧ ˆ e α =D I J∧du I (x) =ω I ∧du I (x). (6.51) In particular, observe that the exterior derivative of J 0 generates precisely the smooth harmonic forms wedged with the exterior derivatives of the vector multiplet scalars. This leads to a particularly simple Ansatz for the three-form potential C (3) = − 1 2 W −1 J ab dt∧e a+4 ∧e b+4 − Z −1 3 dt∧dy 5 ∧dy 6 = −W −1 Z Z 0 dt∧J 0 − Z −1 3 dt∧dy 5 ∧dy 6 , (6.52) 120 for some potential functions, W and Z 3 . The four-form flux is then: F (4) = 1 2 W −1 Z 1/2 e μ α h ω Iab ∂ μ u I −J ab (∂ μ log(WZ 0 /Z)) i e 0 ∧e α ∧e a+4 ∧e b+4 − Z −1 3 Z 1/2 Z Z 0 2 (∂ μ logZ 3 )e 0 ∧e α ∧e 9 ∧e 10 , (6.53) whereω Iab = 2J ac S I c b arethecoefficientsoftwo-formω I . Wehavealsoconsideredaddingexplicit terms proportional to ω I in the Ansatz (6.52) for C (3) , as in (6.47). However, solving the BPS equations eliminated these terms and showed that the simple Ansatz, (6.52), and the ω I -terms that it generates in F (4) , are sufficient. 6.4.4 The BPS equations The BPS equations are given by the vanishing of the gravitino variations, which we write in frames: e M A δψ M = e M A ∂ M ε + 1 4 ω ABC Γ BC ε + 1 288 Γ A BCDE F BCDE − 8Γ BCD F ABCD ε = 0. (6.54) We will also impose the projection conditions (6.15) onε. In particular, we note that the second projection in (6.15) implies that for any anti-self-dual two-form,A, on internal four-fold one has: A ab Γ a+4b+4 ε = 0. (6.55) Assuming that Killing spinor, ε, is time independent, the 0-component of BPS equations gives: e M 0 δψ M = 1 2 Z −1/2 e μ α " (∂ μ logZ) Γ 0α − 1 6 ω Iab ∂ μ u I − Z W (∂ μ log(WZ 0 /Z))J ab ! Γ αa+4b+4 + 1 3 Z Z 0 2 Z Z 3 (∂ μ logZ 3 ) Γ α910 # ε = 0. (6.56) 121 Using the projection conditions and (6.55) to eliminate the ω I -term, reduces this to: e μ α 1 2 Z −1/2 Γ 0α ε " ∂ μ logZ − 2 3 Z W ∂ μ log(WZ 0 /Z) − 1 3 Z 3 Z 2 0 Z 3 ∂ μ logZ 3 # = 0. (6.57) Similarly, the component of (6.54) parallel to the spatial sections ofX gives the equation: Z −1/2 e μ γ h ∂ μ + 1 6 ∂ μ logZ 3 + 1 3 ∂ μ logZ 0 i ε + 1 12 Z −1/2 e μ α Γ γ α ε h 3∂ μ logZ− 2∂ μ logZ 0 −∂ μ logZ 3 i = 0, (6.58) where we have used the projection condition (4.4) combined with the self-duality of the spin- connection, ˆ ω ˆ α ˆ β , onX to eliminate the spin connection terms in (6.58). The component parallel to y 5 or y 6 leads to − 1 6 Z −1/2 e μ α Γ Aα ε h 3∂ μ logZ − 2∂ μ logZ 0 − ∂ μ logZ 3 i = 0, (6.59) for A = 9, 10. As a result of (6.57) – (6.59), we see that Z = (Z 2 0 Z 3 ) 1/3 , W = Z, (6.60) and ε = Z −1/2 ε 0 , (6.61) where ε 0 is a constant spinor. In particular (6.61) is also required by the fact that (6.70) gives the time-like Killing vector. The results of these supersymmetry variations closely parallel the computations for the T 6 compactification. However, there are potentially dangerous extra terms that cancel as result of anti-self-duality of the tensors involved and the identity (6.55). The last set of supersymmetry variations are the ones parallel toM GH and because these are 122 quite non-trivial, we describe them rather explicitly. In the frame direction labelled by c, and using W =Z, we have: 1 2 Z −1/2 e μ α " S Iac ∂ μ u I Γ a+4α + 1 2 (∂ μ log(Z/Z 0 )) Γ c+4α − 1 6 (∂ μ logZ 3 ) Γ c+4 0α 9 10 − 1 6 1 2 (∂ μ logZ 0 )J ab − J ad S I d b ∂ μ u I Γ c+4 0αa+4b+4 + 1 3 (∂ μ logZ 0 )J c a − 2J c d S I d a ∂ μ u I Γ 0αa+4 # ε + 1 4 Z −1/2 Z 1/2 0 e ω cab Γ a+4b+4 ε = 0. (6.62) The last term vanishes because of (6.55) and the anti-self-duality of e ω cab in the indicesa,b. It is tempting to try to eliminate the fifth term in the same manner because of the the anti-self-duality of 1 2 ω Iab =J ad S I d b . This is incorrect because of the skew-symmetrization in all three indices [abc] on the Γ-matrices. To handle the middle line in (6.62), one must first use the second projection condition in (6.15) to write: Γ c+4a+4b+4 ε = abcd Γ d+4 ε, (6.63) The expression now involves Γ 0αd+4 , and to simplify this one uses the projection conditions in (6.15), to obtain: Γ 0d+4 ε = −J d b Γ b+4 ε, (6.64) Finally, one uses either the self-duality of J, or the anti-self-duality of ω I to get rid of the abcd . In this way, one arrives at the identity: e μ α 1 2 (∂ μ logZ 0 )J ab − J ad S I d b ∂ μ u I Γ c+4 0αa+4b+4 ε = e μ α h (∂ μ logZ 0 ) Γ c+4α − 2∂ μ u I J ce J db S I d e Γ b+4α i ε = e μ α h (∂ μ logZ 0 ) Γ c+4α + 2∂ μ u I S I c b Γ b+4α i ε, (6.65) where the last equality holds because of the skew-symmetry of 1 2 ω Ibe =J bd S I d e =−J ed S I d b . Finally, one uses (6.64) to simplify the last terms in (6.62) and collects everything together. 123 ThetermsinvolvingS I c b Γ b+4α εcanceldirectlywithoneanother, andtheothertermsaresimply: 1 12 Z −1/2 e μ α Γ c+4α ε 3∂ μ logZ− 2∂ μ logZ 0 −∂ μ logZ 3 (6.66) which vanishes by virtue of (6.60). Therearesomeimportantmessagescomingfromthisdetailedanalysis. First, thecancellation of the various terms in the BPS equations makes heavy use of the relationship, (6.25) or (6.36), between the Lichnerowicz modes and the harmonic forms. The former arise in the gravitino variation via the spin connections while the latter arise in the Maxwell flux, (6.53). The non- trivial fibration form of the metric, (6.17), using the compensators, is crucial to the correct, and non-singular, appearance of the Lichnerowicz modes and harmonic forms in the gravitino variation, and is thus essential to satisfying the BPS equations. Finally, we have arrived at the primary result of this chapter: the BPS equations are identically satisfied by our Ansatz and there is nothing perturbative about this result: it is exact. 6.5 Equations of motion 6.5.1 The BPS system, equations of motion and M2-brane sources Since the commutator of two supersymmetries generates the Hamiltonian of a system, it follows that the integrability condition of the BPS system should lead to at least some of the equations of motion. In particular, solving the BPS equations means that one has automatically solved at least a subset of the equations of motion. In some contexts, solving the BPS equations actually leads to solving all the equations of motion but this is not true in general. For eleven- dimensional supergravity this was investigated in some detail in [84], where it was shown that 124 the BPS integrability conditions are: E MN Γ N ε − 1 6·3! ∗ (d∗F + 1 2 F∧ F ) P 1 P 2 P 3 (Γ M P 1 P 2 P 3 − 6δ P 1 M Γ P 2 P 3 )ε − 1 6! dF P 1 P 2 P 3 P 4 P 5 (Γ M P 1 P 2 P 3 P 4 P 5 − 10δ P 1 M Γ P 2 P 3 P 4 C 5 )ε = 0, (6.67) where E MN ≡ h R MN − 1 12 (F MP 1 P 2 P 3 F N P 1 P 2 P 3 − 1 12 g MN F 2 ) i . (6.68) It follows that, ifF satisfies its Bianchi identity and its equation of motion, then one must have: E MN Γ N ε = 0. (6.69) If ε is a supersymmetry then the vector defined by: K M = ¯ ε Γ M ε, (6.70) is necessarily a Killing vector (see, for example, [84]). Moreover, suppose that ε satisfies the projection condition: 1 l + Γ 0AB ε = 0, (6.71) for some A,B. Then, by inserting Γ 0AB into the right-hand side of (6.70) and acting on both ε and ¯ ε, one can easily show that: K C = ±K C ⇔ Γ C Γ 0AB = ± Γ 0AB Γ C . (6.72) From this, and the conditions (4.3), it follows that the only non-zero component ofK A is K 0 =ε † ε, and thus this Killing vector is necessarily time-like. Then E MN = 0 for all M,N. In other words, the BPS equations, the Maxwell equations and the Bianchi identities necessarily imply the Einstein equations. OurprojectionconditionsimplythatK M istime-likeandso,havingsolvedtheBPSequations, 125 it suffices to check the Maxwell equations and Bianchi identities. Given our Ansatz, (6.52), the latter are automatically satisfied, and so it remains to examine the Maxwell equations, which we will do in this section. Oneshouldnotethattheproofin[84]wasdonepurelyfortheeleven-dimensionalsupergravity action, without explicit brane sources. We are going to assume the corresponding result with M2-brane sources. While this has not been explicitly proven, what matters is how the BPS integrability condition incorporates the inclusion of brane sources in a supersymmetric action. The structure of those integrability conditions must simply add extra source terms to each of the terms in the first line of (6.67). Thus, even with supersymmetric M2-brane sources, solving the BPS equations and satisfying the Bianchi identities and Maxwell equations forF, imply that the Einstein equations will also be satisfied. AnotherissuethatwillariseisthesmearingoftheM2-braneswithinthefull,eleven-dimensional solution. In a Calabi-Yau compactification, wrapping M2-branes on 2-cycles inY, gives rise to electric-charge sources in the effective five-dimensional theory. The electric field lines of these sources can only extend to infinity in the five-dimensional space-time,X, and so, on scales much larger than the compactification scale, the electric potentials fall off as r −2 in the space-time. If one probes the solution on scales less than the compactification scale, one should expect to see deviations from the effective field theory and see details of the M2-brane distribution in the full eleven-dimensional geometry. For torus compactifications, the issue of effective field theories can be obviated through con- sistent truncation. If the compactification manifold has a transitive symmetry group, then it is always a consistent truncation if one restricts to all the fields that are independent of the com- pactification manifold. That is, the reduced theory in lower dimensions is not merely an effective theory, it is actually a consistent truncation in that solving the lower-dimensional equations of motionyieldsanexact(asopposedtoapproximateoreffective)solutiontothehigher-dimensional equations of motion. To incorporate branes in such a consistent truncation, they must be uniformly smeared over 126 the compactification directions that are transverse to the branes. This will preserve the transitive symmetry on all the compactification directions and thus incorporate such smeared brane sources within the consistent truncation. For the M2-branes wrapping aT 2 inside aT 6 compactification, the branes can be uniformly smeared over the transverse T 4 . In a flat spatial base,B, this leads to a purer −2 behavior and the brane sources can be concentrated into a delta-function onB. For non-trivial compactifications on manifolds without transitive symmetries, likeM GH ×T 2 , one must necessarily return to either using an effective field theory that is valid on scales much larger that the compactification scale, or, if one seeks a solution to the eleven-dimensional system, one must make choices of brane distributions on the compactification manifold and solve the eleven-dimensional equations with those brane sources. We now investigate these issues in a little more detail and, for simplicity, we will takeB to be flatR 4 in the rest of this chapter. 6.5.2 The Maxwell equations Since we are using an electric Ansatz for the fluxes, (6.52), one hasF∧F = 0, and the left-hand side of the Maxwell equations becomes: d∗F (4) = vol 4 ∧ h 2Z 0 J + (2∂ μ Z 0 ∂ μ u I +Z 0 2u I )ω I +Z 0 ∂ μ u I ∂ μ u J D J ω I i ∧vol T 2 − Z 0 Z 3 2/3 2Z 3 vol 4 ∧vol 0 4 , (6.73) where2 is a standard Laplacian onB =R 4 , and vol 4 and vol 0 4 are the volume forms onB and M GH . Note also that, because of the compensators, the exterior derivative on ω I appearing in F (4) (see (6.53)) has been promoted toD J ω I . One can easily evaluate this explicitly and the results depends on the details of the harmonic forms: D J ω I = " ∂ a 1 V ∂ 3 K I K J V −δ IJ K I !!# Ω (a) − + ∂ 3 K J V !! ω I + ~ ∇ K I V ! · ~ ∇ K J V ! (J− Ω (3) − ). (6.74) 127 The electrostatic potential, Z 3 , behaves exactly as it does in the T 6 compactification: It is a harmonic function and decouples from other scalars. We will therefore make the standard, spherically symmetric choice: Z 3 = 1 + Q 3 r 2 . (6.75) The interesting new features are associated withM GH and its modulus, and so we focus on these. First, for small fluctuations ofu I about the their stationary values, we can neglect the terms that are quadratic in∂ μ u I in the Maxwell equations. SinceJ and theω I are linearly independent and provide a basis of harmonic (1, 1)-forms, one obtains a set N equations (to linear order in fluctuations): 2Z 0 = 0, 2∂ μ Z 0 ∂ μ u I +Z 0 2u I = 0, I = 1...N− 1, (6.76) or 2Z 0 = 0, 2(Z 0 u I ) = 0 ⇒ u I = Z I Z 0 , 2Z I = 0. (6.77) Thus the moduli are ratios of harmonic functions. If we further impose spherical symmetry one obtains 4 : Z 0 = 1 + Q 0 r 2 , Z 0 u I = 1 + Q I r 2 ⇒ u I = r 2 +Q I r 2 +Q 0 . (6.78) As one would expect, these results correspond precisely with the effective five-dimensional field theory that emerges from a compactification when the manifold,Y, is small. Indeed, to arrive at this result we merely assumed that the fluctuations in the moduli, u I , were small. Note also that the moduli undergo the expected attractor behavior between r = 0 and r =∞. Moving away from slowly varying moduli and effective field theory, we can consider (6.73) in general and see what it implies for eleven-dimensional BPS solutions. 4 One can scale the harmonic functions by a constant and in this way one can give the Z I andu I an overall scale. We have chosen to make the moduli, u I , dimensionless. 128 Onecandefineathree-formchargedensity,λ, astheright-handsideoftheMaxwellequations: ∗d∗F (4) ≡ −λ. (6.79) In principle, one can then allow any choice for the fields Z 0 and u I , and this will lead to a BPS solution to the equations of motion for some distribution of M2 branes. In practice, we want to find BPS solutions for rather more physical choices of brane distributions. The obvious choice is to consider the solution to some form of five-dimensional, effective field theory and determine the corresponding distributions of M2 branes in eleven-dimensional supergravity. The best, most canonical choice is to render∗λ topologically trivial onM GH . That is, we take the right-hand side of (6.73) and project onto all the harmonic forms 5 , which will include extracting the harmonic pieces, proportional to J and each ω I , fromD J ω I . If one sets the coefficients of the harmonic projections of (6.73) to zero, the result will be N equations for Z 0 and the u I and, unlike (6.76), these equations will include the ∂ μ u I ∂ μ u J terms. We will refer to this as the topological solution. Note that, depending on the harmonic content ofD J ω I , it may differ from the solution to (6.76) at the non-linear level. Solving such a homogeneous system really means that one is choosing aδ-function source for thebranechargesatsomelocationinB. In(6.78), thebranesarealllocatedinaδ-functionsource at r = 0. Away from such a δ-function source, the topological solution means that∗λ is exact. As a result, any integral of∗F (4) over a Gaussian surface that excludes theδ-function source will be zero, and thus the M2-brane charge density represented by∗λ is entirely dipolar outside the δ-function source. In addition, the projection of∗λ onM GH is also exact. Consequently, for any point inB that lies outsideδ-function source, the charge distribution on theM GH fibered above that point has no net charge on any cycle inM GH and so, once again, the charge distribution is dipolar. For torus compactifications, or any other compactification manifold with a transitive isometry group, theM2-brane chargedensity,λ, canbesmeared sothat thedipolar distribution isreplaced 5 Here we mean harmonic onM GH , neglecting any dependence of the moduli on the space-time. 129 by its average, and hence is exactly zero outside the δ-function source. The cost of using more general compactification manifolds is that the brane distribution is typically non-zero but entirely dipolar on the compactification fibers outside the δ-function source. This means that if one looks on scales much larger than the compactification, the dipolar distributions average to zero, and the non-zero brane charges are only localized at the δ-function source inB, as required by effective field theory. Zooming in on the compactification scale, one sees how the supersymmetric topological solution resolves into dipolar brane distributions outside the source. More generally, given that the M2-brane charge densities do not necessarily vanish outside the δ-function sources inB, one can also consider other supersymmetric, physical brane distributions that will reduce to something similar to the results one gets from the topological solution. For example, rather than concentrating the brane charge entirely in aδ-function inB, one can spread the brane charge out inB on the scales comparable to that of the compactification and, in so doing, rearrange the dipole densities inM GH . If one wants to go beyond the topological solution, there are many choices and we will not dwell upon them here. We will examine this further by computing an explicit example. 6.6 An example: Eguchi-Hanson We now consider compactifying on the two-centered Gibbons-Hawking metric with unit charges, commonly known as the Eguchi-Hanson (EH) metric. This metric can be obtained as a hyper- Kähler resolution of the conical singularity ofC 2 /Z 2 and depends on the blow-up parameter a, which is the only modulus. One can also think of the EH metric as a “local model” of a K3 metric near a single, isolated 2-cycle. The harmonic (1, 1)-form, ω, is dual to the blown-up cycle and, together with the complex structure,J,formsabasisofthecohomology. Moreover,sinceJ∧ω = 0,theMaxwellfieldsarising from this compactification have similarly trivial intersection and thus we get simpleU(1)×U(1) gauge fields. Including the Maxwell field from the T 2 factor, we obtain the standard “STU” 130 effective field theory in five dimensions. Indeed, the Eguchi-Hanson compactification closely parallels the role of the T 4 (or, more precisely, theR 4 ) in the T 6 compactification discussed in Chapter 4. For the T 6 compactification we can make this parallel more precise by noting that J = udy 1 ∧dy 4 + u −1 dy 2 ∧dy 3 , ω = udy 1 ∧dy 4 − u −1 dy 2 ∧dy 3 . (6.80) defines a self-dual complex structure and an anti-self-dual harmonic form on T 4 . Moreover, one also has J∧ω = 0 and, from the perspective of effective field theory, the compactifications on T 6 andM EH ×T 2 yield the same five-dimensional theory. 6.6.1 Complex coordinates and trivializing the compensators The two-centered EH metric has a lot more symmetry that a general GH metric and so, rather than use the coordinates and frames described in Section 6.2, we will use complex coordinates, (z 1 ,z 2 ), adapted to the SU(2)×U(1) symmetry and Kähler structure of the metric: z 1 = ρ cos θ 2 ! e i 2 (ψ+φ) , z 2 = ρ sin θ 2 ! e i 2 (ψ−φ) . (6.81) We will also see that this choice of coordinates greatly simplifies the compensators and related structure that we described in Section 6.4.1. The Kähler potential is given by K(ρ 2 ) = q ρ 4 +a 4 +a 2 log ρ 2 √ ρ 4 +a 4 +a 2 ! , (6.82) and the Kähler form is J = i 2 ∂ ¯ ∂K(ρ 2 ) = ρ 3 √ ρ 4 +a 4 dρ∧σ 3 + q ρ 4 +a 4 σ 1 ∧σ 2 , (6.83) 131 where the Pauli matrices are σ 1 = 1 2 (sinψdθ− sinθ cosψdφ), σ 2 =− 1 2 (cosψdθ + sinθ sinψdφ), σ 3 = 1 2 (dψ + cosθdφ). (6.84) The metric is given by e ds 2 = 1 q 1 + a 4 ρ 4 dρ 2 +ρ 2 σ 2 3 +ρ 2 s 1 + a 4 ρ 4 σ 2 1 +σ 2 2 , (6.85) and harmonic two-form may be written: ω = − a 2 ρ 3 (ρ 4 +a 4 ) 3/2 dρ∧σ 3 + a 2 √ ρ 4 +a 4 σ 1 ∧σ 2 . (6.86) We will find it convenient to identify the dimensionless modulus, u, as u ≡ a 2 /Λ 2 h , (6.87) where Λ h is a length scale onM EH . This length scale is that of the (compact) homology cycles and, for consistency, it must be much less than any cut-off, Λ ∞ , that one uses to regulate fields at infinity in the EH manifold. One can easily check that ∂J ∂u = Λ 2 h ∂J ∂a 2 = Λ 2 h ω. (6.88) It follows that the Lichnerowicz mode coming from the Kähler deformation of the metric gen- erated by ω is identically equal the derivative of metric with respect to a 2 . The compensator fields are therefore trivial: B i ≡ 0 andD = Λ 2 h ∂ a 2. As we noted at the end of Section 6.4.1, the “curvatures,” Y i IJ , vanish identically and so one might anticipate from this “flatness” that one should be able find a way to trivialize theB i I . The complex coordinates, (6.81), achieve this for the EH metric. 132 Finally, the derivative of ω is simply Λ −2 h Dω = ∂ a 2ω = − ρ 3 (ρ 4 − 2a 4 ) (ρ 4 +a 4 ) 5/2 dρ∧σ 3 + ρ 4 (ρ 4 +a 4 ) 3/2 σ 1 ∧σ 2 . (6.89) It is easy to verify that this has a potential: Dω = d y η, where η ≡ Λ 2 h ρ 4 2(ρ 4 +a 4 ) 3/2 σ 3 . (6.90) The 1-form,η, is “compact” in that it has finite norm and vanishes at infinity. ThusDω is indeed cohomologically trivial on the EH space. Note that these statements about exactness apply only within the cohomology of the EH space: If one replaces d y by d and allows a to become a function of x, then (6.90) is no longer true. Finally, we note the major difference between the harmonic forms onT 4 and the EH manifold. Observe that (6.80) implies u∂ u ω = J, (6.91) which should be contrasted with (6.89) and (6.90). The fact that ∂ u ω is cohomologically trivial in Eguchi-Hanson and non-trivial onT 4 leads to a very significant modification of the differential equation forZ 0 and thus has a significant impact on the form of the solution on the EH manifold compared to the T 4 . 6.6.2 The topological solution Since (6.90) implies thatDω is exact, it follows that the harmonic pieces ofd∗F (4) are precisely the first two terms on the right-hand side of (6.73). As a result, the topological solution described in Section 6.5.2 is exactly the linearized solutions given in (6.76)–(6.78), except that we are no 133 longer linearizing the result. We therefore have: a 2 = Λ 2 h Z 1 Z 0 , 2Z 0 = 2Z 1 = 0, (6.92) where we have set I = 1 in (6.78). It follows from (6.90), (6.73) and (6.79) that we have: ∗λ = d Z 0 ∂ μ u∂ μ u vol 4 ∧ η∧ vol T 2 = d " Z 0 ∂ μ u∂ μ u Λ 2 h ρ 4 2(ρ 4 +a 4 ) 3/2 vol 4 ∧ σ 3 ∧ vol T 2 # . (6.93) For the spherically symmetric solutions in (6.78) this reduces to: Z 0 = 1 + Q 0 r 2 , a 2 = Λ 2 h u = Λ 2 h r 2 +Q 1 r 2 +Q 0 , (6.94) with ∗λ = d " 2 (Q 1 −Q 0 ) 2 (Q 0 +r 2 ) 3 Λ 2 h ρ 4 (ρ 4 +a 4 ) 3/2 vol 4 ∧ σ 3 ∧ vol T 2 # . (6.95) There are several things to note about the brane distribution encoded in∗λ. First,∗λ is exact, as it must be, and is thus intrinsically a multipolar distribution. Indeed, the potential for ∗λ falls off as r −6 inB, as befits a higher multipole distribution of charge. In the internal EH manifold, this potential vanishes at ρ = 0, peaks at ρ = 2 1/4 a and falls off as ρ −2 at infinity and so, in the EH manifold, the brane distribution is localized around the non-trivial cycle at a scale set by a. Thus the EH manifold, unlike T 4 , has a non-vanishing∗λ. However, the charge distribution that it represents involves only higher multipoles that localize in a restricted region ofM GH and very close to r = 0 inB. 134 6.6.3 Another effective action The bosonic action for the “STU” model has the form: S = 1 2κ 2 5 Z √ −gd 5 x R− 1 2 Q IJ F I μν F Jμν −Q IJ ∂ μ X I ∂ μ X J − 1 24 C IJK F I μν F J ρσ A K λ ¯ μνρσλ , (6.96) where I,J = 1, 2, 3 and C IJK =| IJK |. The scalars satisfy the constraint X 1 X 2 X 3 = 1 and metric for the kinetic terms is: Q IJ = 1 2 diag (X 1 ) −2 , (X 2 ) −2 , (X 3 ) −2 . (6.97) The standard way of parametrizing the scalars and the warp factor in the five-dimensional metric is to introduce three scalar fields, Z I , and write: Z ≡ (Z 1 Z 2 Z 3 ) 1/3 , X J ≡ Z Z J . (6.98) We can arrive at another form of the effective action by simply focussing on the scalar kinetic terms. We first eliminate X 3 using X 3 = (X 1 X 2 ) −1 and define the independent dynamical degrees of freedom: A = (X 1 X 2 ) 1/2 and B = (X 2 /X 1 ) 1/2 . On the T 4 , the scalar, A, measures its volume relative to the T 2 , while B determines the relative size of T 2 ×T 2 factors of the T 4 . Comparing with the compactification onM GH ×T 2 ,A is the analog ofZ 0 whileB is the analog of a 2 . Expressing the scalar kinetic term in (6.96) using the fields A and B, we find: S X = Z √ −gd 5 x 3 (∂ μ A) 2 A 2 + (∂ μ B) 2 B 2 ! . (6.99) On the other hand, one can start with the Einstein action in eleven-dimensional supergravity and compute the scalar kinetic terms using the metric Ansatz (6.44). To do this, one must integrate over the non-compactM EH , and this requires a cut-off. We simply choose to integrate ρ from 0 to the scale, Λ ∞ . To simplify this computation we note that we can obtain the form of 135 the scalar kinetic terms by restricting the scalar to be a function of one variable onB. We take this variable to be the radial coordinate, r. The result of this computation is: S scalar = Z √ −gd 5 x 3 (∂ r X 0 ) 2 (X 0 ) 2 + 4a 2 (∂ r a) 2 Λ 4 ∞ +a 4 ! , (6.100) where X 0 =Z/Z 0 . This suggests the following identification between moduli of the our model and the moduli of the STU model 6 : X 0 ↔ A = √ X 1 X 2 , Z 2a∂ r a q Λ 4 ∞ +a 4 dr = log(a 2 + q Λ 4 ∞ +a 4 ) ↔ logB = 1 2 log X 2 X 1 . (6.101) where we have fixed the constant of integration by taking B = 1 when X 2 = X 1 . One obtains the map Z 0 = q Z 1 Z 2 , a 2 = 1 2 Λ 2 ∞ (B−B −1 ) = 1 2 Λ 2 ∞ Z 2 1 −Z 2 2 Z 1 Z 2 , (6.102) for Z 2 1 ≥Z 2 2 . One can make a similar reduction of the eleven-dimensional Maxwell equations to get the five-dimensional effective action of these degrees of freedom. To do this we evaluated the integral of∗F (4) ∧F (4) in eleven dimensions for our flux Ansatz (6.53) and we find: S Maxwell = Z √ −gd 5 x (∂ r Z 0 ) 2 (Z 0 ) 2 + 4a 2 (∂ r a) 2 Λ 4 ∞ +a 4 + 1 2 (∂ r Z 3 ) 2 (Z 3 ) 2 ! . (6.103) With the help of map (6.102) this can be rewritten as S Maxwell = Z √ −gd 5 x 1 2 3 X I=1 (∂ r Z I ) 2 (Z I ) 2 ! . (6.104) This is exactly what one will get from the action of STU model (6.96) if one substitutes the 6 Obviously, given that we are matching quadratic forms, one can make an arbitrary rotation of the identification we are using. 136 electric Ansatz for five-dimensional Maxwell fields F I = −d(Z −1 I dt). (6.105) The five-dimensional theory implies that the functions, Z I , are harmonic, and for spherically symmetric solutions they can be chosen as in (6.78). The identification (6.102) leads to a different eleven-dimensional solution to the one given in Section 6.6.2 While this attempt at arriving at an effective action looks a little more like the T 4 effective action, particularly with the form of Z 0 in (6.102), it is of more limited utility than the solution in Section 6.6.2. This is because we already have requirement Z 1 ≥ Z 2 and we must also keep a Λ ∞ for consistency. This means one must require B ∼ 1, or Z 1 Z 2 ≈ 1. Similarly, the identification (6.102) means that the coefficients ofJ andω in∗λ do not vanish, and so there are non-trivial monopolar charge sources smeared onB. This smearing inB will only remain small if Z 1 Z 2 ≈ 1. With such restrictions we are essentially back in the linearized regime in which the fluctuations of u must remain small. Here we have taken an obvious but rather ‘ad hoc’ procedure to arrive at an effective action andasolution. Theissuewithsuchagenericapproachisthatitwilltypicallyhavealimitedrange application and lead to non-trivial smearing of charge sources in the space-time. In contrast, the topological solution described in Sections 6.5.2 and 6.6.2 produces solutions whose charge distribution is intrinsically dipolar on theY fibers outside the desired space-time sources and that remains valid for generic solutions of the underlying equations of motion in the effective field theory. 6.7 Conclusions The standard microstate geometries consider geometric transitions in the space-time directions while the topology and geometric dynamics of the compactification manifold does not play much role. However, to describe a more sophisticated states, like W-branes, one has to investigate the 137 dynamics of the internal degrees of freedom at the same level of detail as we have investigated the space-time dynamics. This means that we must replace the toroidal compactification with non-trivial Calabi-Yau manifolds whose homology cycles can blow up or blow down without the volume becoming singular. In this chapter we have used a Gibbons-Hawking ALE space,M GH , as a local model of a K3 surface and we have found a complete solution to the eleven-dimensional supersymmetry conditions (BPS equations) for a compactification onM GH ×T 2 that involves purely electric charges. We also required that the supersymmetries have exactly the same structure (with the same projectors) as the three-charge black hole in five dimensions. This solution to the BPS equations requires the compactification manifold to be fibered over the space-time base using the compensator fields onM GH . Solving the BPS equations will generically only solve a subset of the equations of motion. We argued, following [84], that solving the Bianchi identities and the Maxwell equations in addition to the BPS equations, even in the presence of brane sources, implies that the Einstein equations are satisfied. We therefore examined the Maxwell equations. This also highlights a physical issue with non-trivial Calabi-Yau compactifications: In toroidal compactifications one smears the brane distributions over the compactification directions that lie transverse to the branes. However, a generic Calabi-Yau manifold does not possess a set of transitive symmetries transverse to the branes and so there is no canonical definitions of a “uniform” distribution of branes in such transverse directions. Instead one must examine choices of solutions and the brane distributions to which they lead. The most canonical is our “topological choice”, in which the solutions are chosen so as to eliminate all the sources proportional to the harmonic forms on the internal manifold. This is the most natural choice for several reasons. First, this is the choice motivated by the effective five-dimensional field theory that emerges from the topology of the internal manifold. Secondly, making the charge density exact on the internal manifold means that it carries no net charge and is therefore intrinsically dipolar (outside any explicit sources in the space-time). On scales much larger than the compactification scale, the fields sourced by 138 such a distribution fall off very rapidly. Indeed, for theM EH ×T 2 solutions we found that this dipolar distribution localized around the non-trivial cycle. 139 Chapter 7 Tidal stresses and energy gaps in microstate geometries 7.1 Motivation The study of five-dimensional microstates considered in the chapter 4 has reached several goals of the microstate geometry program. In particular, microstate geometries provide a mechanism withing the classical supergravity to resolve the horizon structure of the black hole and replace it with smooth horizonless geometry which has the same macroscopic charges as the black hole. The“bubbledgeometries”cruciallyrelyonChern-Simonsinteractionandthenon-trivialtopology of spacetime which allows one to replace electric charges with cohomological fluxes. However, five-dimensional microstates also have a few issues. First, the early solutions of [85,86] carried angular momentum that was a large fraction of the maximally allowed value for corresponding black hole. Second, the CFT dual of the solutions with multiple Gibbons-Hawking centers still remains unknown, although the CFT dictionary has been worked out for two-centered case. Third, it is unclear how typical these microstates are, and whether one can find enough of them to account for the black hole entropy. One of the properties of bubbled geometries is that they can closely approximate the region 140 outside of a black hole. This requires a long BTZ-like AdS 2 throat which can be obtained by takingascalinglimitofmulti-centeredsolution,whenGibbons-Hawkingcentersbecomearbitrary close to each other. Unfortunately, the minimal number of GH centers required for scaling is three, which makes the CFT interpretation obscure. The new approach to constructing microstates in six dimensions has been proposed recently, and it is based on adding the momentum excitations to a certain two-charged seed solution. This new class of solutions was called “superstratum”, and it has a clear interpretation in terms of dual D1-D5 CFT. By now several examples of these solutions have been obtained both in IIB supergravity [87–94] and in M-theory [95]. The first microstate geometry of rotating, su- persymmetric D1-D5-P BMPV black hole [96] was constructed in [91,94] in which the angular momentum can take arbitrary small values. It has a decoupling limit of the form AdS 3 ×S 3 and the black-hole-like AdS 2 throat that caps off smoothly above the would-be horizon. The depth of the throat is controlled by angular momentum, j, and it becomes infinitely long in the limit j→ 0. Thus, one can approximate a black hole with arbitrary precision at least at the classical level. Recently the superstratum technology has been also applied to obtaining five-dimensional microstates with low angular momentum [92]. In classical theory the parameters that describe microstates are continuous, and the moduli space is infinite dimensional. To find the actual number of microstates one has to semi-classically quantize them. In the simplest examples, the quantization of angular momentum led to a limit on the supergravity moduli [85,86], and a far more sophisticated, and more general analysis, was obtained in [97,98] by quantizing the moduli space of multi-centered solutions. This showed that the tuning of the angular positions of each cycle was limited by the quantized angular momentum carried by that cycle, thereby limiting the classical depth of microstate geometries. The fact that the throat has a finite depth implies that there is a new energy scale in the dual CFT. Indeed, it means that there is a maximal redshift,z max , between the energy of excitation at bottomofthethroatandtheenergymeasuredatinfinity. Ifoneplacestheexcitationwithlongest possiblewavelengththatcouldresideatthebottomofthedeepestthroatthenitsredshiftedvalue 141 will determine the minimal energy in CFT. In other words it means that the spectrum of CFT must be gapped. It was one of the triumphs of the microstate geometry program that this energy gap precisely matched that of the lowest energy states of the dual D1-D5 CFT. In this chapter we use geodesics to probe the scaling microstate geometries constructed in [91,93,95], particularly when these geometries are deep enough to access the very low-energy, intrinsically stringy sectors of the dual D1-D5 CFT. We analyze the geodesic deviation for classes of radially infalling, massive particles and, rather surprisingly, we find that such a particle must undergo a stringy transition long before it reaches the cap of a deep, scaling microstate geometry. We also find that, for a scaling microstate geometry of maximal depth whose excitations of the cap have the gap energy,O((N 1 N 5 ) −1 ), the gravitational stress reaches the Planck scale when the probe is encountering a throat depth corresponding to a CFT energy scale ofO((N 1 N 5 ) −1/2 ), which is that of the typical sector of the D1-D5 CFT. This observation is in the sharp contrast with a celebrated fact that there is “no drama at the horizon of a black hole”, and the tidal forces on an infalling observer near the horizon can be made arbitrarily small by making the mass of the black hole arbitrarily large. What is happening is that, even though the presence of the cap produces extremely small curving effects higher up in thethroat, the ultra-relativisticspeeds ofparticles fallingintovery deepthroats greatly magnifies these tiny “curving effects” to the extent that they rip the particle into its constituent strings before the particle gets to the cap. Microstate geometries are supposed to represent individual microstates of black holes and sufficiently typical microstate geometries should behave much like a black hole. Thus microstate geometries should trap particles and those particles should ultimately “scramble” into the microstate structure. The microstate geometries that we analyze here can be written in a relatively simple form as a warped fibration of anS 3 over a three-dimensional space-time,K [93]. For a scaling geometry, it isK that closely approximates the BTZ geometry while the S 3 remains macroscopic and has non-trivial warp factors and fibration connections. These six-dimensional geometries have a singular limit in which theK becomes precisely an extremal BTZ black hole, the S 3 becomes 142 maximally symmetric while the fibration becomes trivial 1 . From this one might be led to believe that S 3 is simply some macroscopic auxiliary space and that most of the interesting physics lies inK. Indeed, it was noted in [93] that the warp factors exhibit only a modest dimple where the microstate structure is localized. However, we will show that stress forces in the sphere directions not only become extremely large in some directions but also rapidly change sign (twice) as the particle crosses the localized microstate structure. Thus the huge stress forces not only arise from capping off of the BTZ geometry but also as a result of the non-trivial fibration and the bumps at the bottom of the throat. 7.2 The microstate geometries 7.2.1 The CFT states and dual geometries We are going to focus on probing geometries that are holographically dual to the pure momentum excitations of the D1-D5 system. The numbers of underlying D1-branes and D5-branes will be denoted by N 1 and N 5 respectively. The ground states of the D1-D5 system can be described by partitioning N =N 1 N 5 into strands of lengths between 1 and N and by the spins of each of the strands (see, for example, [87]). One can then excite the strands using operators in the CFT and the standard set of 1 8 -BPS states are those that remain in the right-moving Ramond ground state but have arbitrary excitations in the left-moving sector. Here we will work with microstate geometries that are dual to coherent superpositions of states assembled from strands of length 1 with spins|00i and|++i, and we only consider the left-moving excitations obtained by acting with (L −1 −J 3 −1 ) on the|00i strands. That is we consider the states: (|++i 1 ) N ++ 1 n! (L −1 −J 3 −1 ) n |00i 1 ! N 00 , (7.1) where N ++ +N 00 =N≡N 1 N 5 and the subscripts on|...i 1 indicate the strand length. 1 It is in this sense that the microstate geometries can be chosen to be arbitrarily close to BTZ×S 3 . 143 Themicrostategeometriesdualtosuchcoherentstateswerefirstpresentedin[91]. Apartfrom the quantum numbersN 1 ,N 5 andn, there are two Fourier coefficients, a andb, that determined the numbers of|++i 1 and|00i 1 strands, respectively. In the supergravity theory, the partitioning of the strands emerges as a regularity condition at the D1-D5 locus and takes the form: Q 1 Q 5 R 2 y = a 2 + 1 2 b 2 , (7.2) where Q I are the supergravity charges and R y is the radius of the common y-circle of the D1 and D5 branes. The supergravity charges are related to the quantized charges via [87]: Q 1 = (2π) 4 N 1 g s α 03 V 4 , Q 5 =N 5 g s α 0 , (7.3) where V 4 is the volume of T 4 in the IIB compactification to six dimensions. In particular, it is convenient to defineN via: N ≡ N 1 N 5 R 2 y Q 1 Q 5 = V 4 R 2 y (2π) 4 g 2 s α 04 = V 4 R 2 y (2π) 4 ` 8 10 = Vol(T 4 )R 2 y ` 8 10 , (7.4) where ` 10 is the ten-dimensional Planck length and (2π) 7 g 2 s α 04 = 16πG 10 ≡ (2π) 7 ` 8 10 . The quantity, Vol(T 4 )≡ (2π) −4 V 4 , is sometimes introduced [99] as a “normalized volume” that is equal to 1 when the radii of the circles in the T 4 are equal to 1. Ifonehasasupergravitymomentumcharge,Q P ,thenitisrelatedtothequantizedmomentum charge (along the y-direction) via: N P = NQ P . (7.5) 7.2.2 The family of metrics While the microstate geometries dual to coherent superpositions of states of the form (7.1) were first given in [91], the metrics were rewritten in a much more convenient form in [93]. In 144 particular, they were recast in terms of an S 3 fibration over a (2 + 1)-dimensional base,K: ds 2 6 = q Q 1 Q 5 Λ F 2 (r) " F 2 (r)dr 2 r 2 +a 2 − 2F 1 (r) a 2 (2a 2 +b 2 ) 2 R 2 y dv + a 2 (a 4 + (2a 2 +b 2 )r 2 ) F 1 (r) du ! 2 + 2a 2 r 2 (r 2 +a 2 )F 2 (r) F 1 (r)R 2 y du 2 # (7.6) + q Q 1 Q 5 " Λdθ 2 + 1 Λ sin 2 θ dϕ 1 − a 2 (2a 2 +b 2 ) √ 2 R y (du +dv) 2 + F 2 (r) Λ cos 2 θ dϕ 2 + 1 (2a 2 +b 2 )F 2 (r) √ 2 R y h a 2 (du−dv)−b 2 F 0 (r)dv i 2 # , where the functions, F i (r), are defined by: F 0 (r) ≡ 1− r 2n (r 2 +a 2 ) n , F 1 (r) ≡ a 6 −b 2 (2a 2 +b 2 )r 2 F 0 (r), F 2 (r) ≡ 1− a 2 b 2 (2a 2 +b 2 ) r 2n (r 2 +a 2 ) n+1 , (7.7) and the warp factor, Λ, is defined by: Λ ≡ v u u t 1− a 2 b 2 (2a 2 +b 2 ) r 2n (r 2 +a 2 ) n+1 sin 2 θ. (7.8) Thecoordinates,uandv, arethestandardnullcoordinates, whicharerelatedtothecanonical time and spatial coordinates via: u = 1 √ 2 (t−y), v = 1 √ 2 (t +y), (7.9) where y is the coordinate around S 1 with y ≡ y + 2πR y . (7.10) The parameters, a and b, made their original appearance in the microstate geometry as Fourier coefficients of the underlying supertube profile and of a charge-density fluctuation. The 145 metric only depends on the RMS values of these profiles, and hence only upon a 2 and b 2 . The quantized angular momenta and momentum charges of this geometry are related to the Fourier coefficients via [91]: J ≡ J L = J R = 1 2 Na 2 , N P = 1 2 Nnb 2 . (7.11) The identity J L = J R ∼ a 2 reflects the fact that the only source of angular momentum is the |++i supertube and that it has a circular profile in anR 2 of the spatialR 4 base geometry. The excitations are created only on the|00i supertube and so the momentum is proportional to nb 2 . It is also useful to note that (7.2) can be rewritten as b 2 a 2 = 2 Q 1 Q 5 a 2 R 2 y − 1 ! = 2 N 1 N 5 Na 2 − 1 ! = N 1 N 5 J − 2. (7.12) We are going to be particularly interested in microstate geometries that closely approximate non-rotating black holes and so they will be “deep, scaling geometries,” with a 2 b 2 . Such microstate geometries will be characterized by b 2 ≈ 2Q 1 Q 5 R 2 y , a 2 b 2 ≈ J N 1 N 5 . (7.13) Note that a 2 b 2 controls the angular momentum of the CFT state compared to the overall central charge of the CFT. 7.2.3 Limits of the metric The metric, (7.6), is asymptotic to AdS 3 ×S 3 at infinity. Indeed, for large r one has: ds 2 6 = q Q 1 Q 5 " dρ 2 ρ 2 − ρ 2 dt 2 + ρ 2 dy 2 # + " dθ 2 + sin 2 θ dϕ 1 − R y a 2 Q 1 Q 5 dt ! 2 + cos 2 θ dϕ 2 − R y a 2 Q 1 Q 5 dy ! 2 #! , (7.14) 146 where ρ≡ (Q 1 Q 5 ) − 1 2 r and we have used (7.9). At the other extreme, r = 0, the metric becomes degenerate for the simple reason that a (boosted)y-circle pinches off. We therefore expand aroundr, retaining only ther 2 terms needed to avoid degeneracy. We find ds 2 6 = q Q 1 Q 5 " dρ 2 − a 4 R 2 y (Q 1 Q 5 ) 2 dt 2 + ρ 2 dy R y + R y b 2 dt 2Q 1 Q 5 ! 2 # + " dθ 2 + sin 2 θ dϕ 1 − R y a 2 Q 1 Q 5 dt ! 2 + cos 2 θ dϕ 2 − dy R y − R y b 2 2Q 1 Q 5 dt ! 2 #! , (7.15) whereρ≡r/a. One should note that becausey/R y has period 2π, there is no conical singularity at ρ = 0. Indeed this geometry is simply that of Mink (1,2) ×S 3 . The geometry thus caps off smoothly at r = 0. Finally, there is the a→ 0 limit, which leads to the extremal BTZ metric with a round S 3 : ds 2 6 = q Q 1 Q 5 " dρ 2 ρ 2 −ρ 2 dt 2 +ρ 2 dy 2 + n R 2 y (dy+dt) 2 # + h dθ 2 +sin 2 θdϕ 2 1 +cos 2 θdϕ 2 2 i ! , (7.16) where ρ≡ (Q 1 Q 5 ) − 1 2 r. 7.2.4 Some comments on curvatures and the supergravity approxi- mation For supergravity to be a valid description of a microstate geometry, the scales of the essential geometric features, and particularly the curvature (length) scale, should be much larger than the Planck length. For theT 4 compactification of IIB supergravity, the scale of the T 4 must also be larger than the Planck scale: Vol(T 4 ) > ` 4 10 . (7.17) 147 Furthermore, for the six-dimensional supergravity to be a valid description, the six-dimensional geometry, and R y in particular, should also be larger than the scale of the T 4 . We will typically find that geometric details and tidal stresses can be expressed as a multiple of some combination of ` 10 and Vol(T 4 ) and we will generically refer to these scales as the Planck/compactification scale. When possible, we will use (7.17) to relate everything to the ten-dimensional Planck scale. The AdS 3 ×S 3 metric is, of course, maximally symmetric and the BTZ metric is that of AdS 3 divided by a discrete group and so the curvature is that of AdS 3 . It follows that the two limits, (7.14) and (7.16), have radii of curvature, `, given by: ` −2 = 1 √ Q 1 Q 5 ∼ √ 2 bR y , a→ 0. (7.18) Thus, the curvatures vanish uniformly for large b. The general metric, (7.6), is obviously much more complicated. However, for small a b , there is a long BTZ throat and the region around r = 0 caps off smoothly in Mink (1,2) ×S 3 while the scale of the S 3 remains macroscopic. One therefore expects that the curvatures remain small and the supergravity approximation to remain valid all the way down to r = 0. Indeed, we have computed the Riemann invariant, and we find that it is everywhere regular and, to leading order in inverse powers of b, it behaves as: R ρσμν R ρσμν ∼ P (r, sin 2 θ) ((r 2 +a 2 ) 2 −a 2 r 2 sin 2 θ) 5 1 b 2 R 2 y . (7.19) whereP (r, sin 2 θ) is a complicated polynomial of degree 20 inr and degree 4 in sin 2 θ. Thus the Riemann invariant of the general metric, (7.6), exhibits the same overall scaling, (7.18), inbR y as the AdS 3 ×S 3 and the BTZ metric. In particular, the curvature invariant becomes vanishingly small for large b. It therefore appears, at least from the perspective of smoothness and curvature, that super- gravity remains valid for any value of a b , including the limit a b → 0. 148 Perhaps the most important geometric feature is the cap and this becomes manifest atr∼a. One has, from (7.11), that a 2 = 2J N = 2` 8 10 J Vol(T 4 )R 2 y , (7.20) where we have used (7.4). Thus, for J∼O(1), a is essentially at the Planck/compactification scale. This seems to be at the edge of the supergravity limit, however, a itself is not a physical scale in the metric. Indeed (7.6) implies that the scale of the cap is approximately (Q 1 Q 5 ) 1/4 Z r=a r=0 √ Λdr √ r 2 +a 2 ∼ (Q 1 Q 5 ) 1/4 , (7.21) which is the horizon scale, and so the cap is always macroscopic. It is also very instructive to examine the typical size of the fluctuations in the size of the S 3 at various points in the BTZ throat and at the cap. If one sets r =a (1−α) b α for some 0<α< 1 then, for ba, one finds that the circumference of a great circle defined by θ is given by 2 (Q 1 Q 5 ) 1/4 Z θ=π θ=0 dθ √ Λ ∼ 2π(Q 1 Q 5 ) 1/4 1 − 1 8 a 2 b 2 ! α ! , (7.22) The deviation from the round circumference has magnitude π 4 (Q 1 Q 5 ) 1/4 a 2 b 2 ! α ∼ ` 2 10 J α (N 1 N 5 ) 1 4 −α (Vol(T 4 )) 1 4 , (7.23) This is below the Planck length for α > 1 4 and macroscopic for α < 1 4 . Indeed, for r = a the circle has circumference 2 (Q 1 Q 5 ) 1/4 Z θ=π θ=0 1− 2 −(n+1) sin 2 θ 1/4 dθ. (7.24) The deviation is once again of orderO((Q 1 Q 5 ) 1/4 ) and therefore of similar scale to the horizon of the corresponding black hole. Inthissense, themetric, (7.6), isextremelyclosetothatofBTZ×S 3 untilclosetothebottom 149 of the throat, at which point, the cap and all the essential structural features grow much larger than Planck/compactification scale and indeed have scales comparable to that of the horizon of the corresponding black hole. One therefore expects that supergravity should accuately capture the essential physical details of this geometry and, most particularly, the cap and the fluctuations caused by the microstate structure. 7.3 The energy gap, red shifts and dispersion relations Because of the twisted sectors of the D1-D5 CFT, the excitations of the D1-D5 CFT can have energies that are fractionated relative to the energy scale, R −1 y , suggested by the size of the common circle on the D1’s and D5’s. Indeed, for strands of length k, the lowest energy mode has an energy∼k −1 , in units ofR −1 y . The maximally-twisted sector has strands of length N 1 N 5 and so it gives rise to the lowest energy excitations of the D1-D5 theory, with E gap = 1 N 1 N 5 . (7.25) This maximally-twisted sector is extremely important because its extremely small energy gap leads to vast degeneracies of states and these provide a dominant contribution to the entropy of the BPS black hole [10]. 7.3.1 Red shifts and E gap One of the triumphs of the microstate geometry program was to show that this sector of the CFT can be accessed using deep, scaling microstate geometries. The original argument was simple [85,86]: scaling geometries could not be made arbitrarily deep because the angular momentum of the geometry was quantized. Setting the angular momentum exactly to zero makes the throat infinitely long, and the geometry becomes singular. As a result, the lowest angular momentum isO(~) and so there is a maximum depth. The energy gap then emerged from looking at the longest wavelength mode that could reside at the bottom of such a throat and then computing 150 its energy including the huge redshift between the bottom and the top of the throat. The result was (7.25). A much more general argument was given in [97,98], where the moduli space of multi-centered microstate geometries was semi-classically quantized. If one looks at the coefficient ofdt 2 in the metric (7.15), one immediately notices the huge red- shift factor of a 2 Ry (Q 1 Q 5 ) 3/4 , where we have taken the square-root to get the scaling of the proper time. Onecanthenuseargumentsexactlyparalleltothoseof[85,86]toarriveat(7.25). However, given some of the remarkable properties of the metric (7.6), we will make a slightly more sophisticated analysis using the dispersion relation for a massless scalar field in (7.6). 7.3.2 Dispersion relations In [93] it was shown that the massless wave equation was separable in the background (7.6). In particular, for a a generic mode, Φ, of the form Φ = K(r)S(θ)e i √ 2 Ry ωu+ √ 2 Ry pv+q 1 ϕ 1 +q 2 ϕ 2 . (7.26) the massless wave equation reduces to: 1 r ∂ r r(r 2 +a 2 )∂ r K + a 2 (ω +p +q 1 ) 2 r 2 +a 2 − a 2 (ω−p−q 2 ) 2 r 2 ! K (7.27) + b 2 ω (2a 2 p +F 0 (r)[2a 2 (ω +q 1 ) +b 2 ω]) a 2 (r 2 +a 2 ) K = λK, 1 sinθ cosθ ∂ θ sinθ cosθ∂ θ S − q 2 1 sin 2 θ + q 2 2 cos 2 θ ! S = −λS, (7.28) for some eigenvalue λ. The second equation has regular solutions based on Jacobi Polynomials of cos 2θ provided that λ = `(` + 2). (7.29) Indeed, the solution to the eigenvalue problem, (7.28), is simply provided by the harmonic modes on a round S 3 of unit radius. 151 Ifb were zero then the first eigenvalue problem, (7.27), would reduce to finding the harmonic modes on AdS 3 . The Laplacian for this is simply: 1 r ∂ r r(r 2 +a 2 )∂ r K + a 2 (ω +p +q 1 ) 2 r 2 +a 2 − a 2 (ω−p−q 2 ) 2 r 2 ! K = λK. (7.30) The modes of this Laplacian are non-trivial fluctuations of ρ =r/a, and hence vary significantly onscalesr∼a. Asnotedin(7.21), suchmodesactuallyhaveawavelengthoforderO((Q 1 Q 5 ) 1/4 ), which is also the scale of the S 3 . The modes we are interested in are precisely these long- wavelength modes, and their counterparts on the S 3 . We therefore want K(r) to be either approximately constant or a slowly varying function of ρ =r/a. The Laplacian (7.30) blows up at r = 0 but this is a standard artifact of using polar coordi- nates. Near r = 0, (7.27) reduces to: 1 r ∂ r r∂ r K − (ω−p−q 2 ) 2 r 2 K = ˜ λK. (7.31) where we have simply put all the constants into a new eigenvalue of ˜ λ. This is just polar form for the Laplacian onR 2 , as one should expect from the capping off in Mink (1,2) . Regularity requires that (ω−p−q 2 ) = m∈Z, (7.32) and that K(r)∼r m as r→ 0. The constant mode, of course, corresponds to m = 0. Returning to the original issue, we want to consider modes that are localized around r = 0 but have the longest wavelength, this means that we want to solve (7.27) near r = 0 with K(r) either constant or slowly varying in the cap. If K(r) is not constant it must satisfy (7.32) and vanish according to K(r)∼r m as r→ 0. We therefore drop the first and third terms in (7.27) and set r = 0 in the remaining part. Taking K(r)∼ constant (m = 0) and dropping the overall 152 factor of K, one obtains the dispersion relation: (ω +p +q 1 ) 2 + 2b 2 a 4 ω a 2 (p +q 1 ) + (a 2 + 1 2 b 2 )ω = `(` + 2), (7.33) The simplest non-trivial mode on the S 3 is S(θ) = cos 2θ, which has q 1 = q 2 = 0, ` = 2 and henceλ = 8. We can eliminatep using (7.32) and we can takeq 1 ,q 2 small and for b a large, (7.33) reduces to ω = a 2 b 2 μ ∼ μJ N 1 N 5 , (7.34) where μ is some number of order q `(` + 2)∼O(1) and where we have used (7.13) to arrive at the second identity. More generally, for K(r) varying slowly in the cap, the constant μ is modified by the eigenvalue, ˜ λ, in (7.31), but this is also a number of order 1. Thus we obtain a result of the form (7.34) for modes of wavelength of orderO((Q 1 Q 5 ) 1/4 ) whether they lie on the sphere or in the cap. The deepest scaling geometries have J = 1 and so we arrive at ω ∼ μ N 1 N 5 . (7.35) This implies that for such geometries, the gap energy is given by (7.25), as one might have expected. However, we have also learnt something else from this analysis of the dispersion relations: Regularity of modes in the microstate geometry imposes level-matching on the dual CFT state. Specifically, the quantum numbersp andω in (7.26) represent the left-moving and right-moving energies of the modes moving around the y-circle. As we saw in (7.32), regularity of the modes at r = 0 requires: (ω−p)∈Z. (7.36) This is precisely the level-matching condition: L 0 − ¯ L 0 ∈Z on CFT states. Our computation, and in particular (7.35), shows the quantum numbers p and ω can both fractionate in units of 153 (N 1 N 5 ) −1 but their difference must be an integer, exactly as in the dual CFT. Level-matching is, of course, an expression of world-sheet angular momentum and so its integer quantization is hardly surprising. However, here we see it emerging from regularity at the cap of a microstate geometry. This observation goes some way to explaining why finding fractionated modes in BPS mi- crostate geometries has proven so elusive: the BPS condition requires the right-moving sector to be in its ground state while regularity of the solution requires level-matching. This means that smooth BPS microstate geometries cannot see anything other than collective excitations with integer-valued left-moving energies. Energy fractionation is only visible in regular solutions if one looks at non-BPS excitations, like those of (7.26). It is also interesting to note that this observation meshes nicely with the results of [89], which studied microstate geometries with an orbifold singularity. These geometries did indeed access fractionated BPS excitations on the left-moving sector but only in combinations that respected level-matching. 7.4 Geodesics and probes 7.4.1 Radially infalling geodesics Geodesics in the metric (7.6) were studied in [93], where it was shown that there is a conformal Killing tensor and hence there is an additional quadratic integral of the motion for null geodesics. Here we are going to be concerned with a simple class of time-like geodesic probes. For simplicity, we will look at “equatorial geodesics” at θ = 0 and θ = π 2 and hence dθ dτ = 0. One can easily check that such a restriction is consistent with the geodesic equations because of the symmetries of the metric under θ→−θ andθ→π−θ. Atθ = 0 the coordinate, ϕ 1 , degenerates and so we will have dϕ 1 dτ = 0. Similarly, at θ = π 2 the coordinate, ϕ 2 , degenerates and we have dϕ 2 dτ = 0. Recall that the warp-factor, Λ, is given by (7.8) and so the geodesics with θ = π 2 “see the bump” more sharply, while the geodesics for θ = 0 see the bump less strongly. 154 The isometries guarantee the following conserved momenta 2 : L 1 = K (1)μ dx μ dτ , L 2 = K (2)μ dx μ dτ , P = K (3)μ dx μ dτ , E = K (4)μ dx μ dτ , (7.37) where the K (I) are the Killing vectors: K (J) = ∂ ∂ϕ J , K (3) = ∂ ∂v and K (4) = ∂ ∂u . The standard quadratic conserved quantity coming from the metric is: g μν dx μ dτ dx ν dτ ≡ −1, (7.38) which means that τ is the proper time measured on the geodesic. One can now use dθ dτ = 0 and (7.37) to determine all the velocities with the exception of dr dτ , however, as usual, this can be determined, up to a sign, from (7.38). Since we want to consider infall, we want dr dτ < 0. To remove all the centrifugal barriers and enable the geodesic to fall from large values of r down to r = 0, one must take: L 1 = 0, L 2 = 0, P = E. (7.39) One should note that at infinity this means that du dτ = dv dτ = − E √ Q 1 Q 5 r 2 ⇒ dt dτ = − E √ 2Q 1 Q 5 r 2 , dy dτ = 0. (7.40) Thus the particle has no y-velocity at infinity and, for standard time-orientations ( dt dτ > 0), one must have E < 0. (7.41) 2 As usual with geodesics, these quantities are “momenta per unit rest mass,” and so their dimensions must be adjusted accordingly. 155 Using (7.39), one finds that (7.38) can be reduced to dr dτ 2 = (2a 2 +b 2 F 0 (r)) a 2 √ Λ E 2 − (r 2 +a 2 ) R y q a 2 + 1 2 b 2 √ Λ , (7.42) where Λ is the warp factor, (7.8), andF 0 (r) is defined in (7.7). Note that forθ = 0 one has Λ = 1 while for θ = π 2 one has Λ =F 2 (r). One should also note that in arriving at this expression we have used dθ dτ = 0 and so it is only valid for geodesics at constant θ. For infall, one takes the negative square-root. At r = 0, this becomes dr dτ 2 = 2Q 1 Q 5 a 2 R 2 y E 2 − a 2 √ Q 1 Q 5 , (7.43) where we have used (7.2). There is thus no centrifugal barrier, and, to leading order, we have: dr dτ r=0 ∼ √ 2Q 1 Q 5 aR y E. (7.44) For r,ba, one has: dr dτ 2 = 2 + nb 2 r 2 E 2 − √ 2r 2 R y b , (7.45) which is precisely what one obtains for similar geodesics in the BTZ metric (7.16) if one uses (7.2) with ba. Note that if these radial geodesics come to a halt at r =r ∗ a then E 2 = √ 2r 4 ∗ R y b 2r 2 ∗ +nb 2 . (7.46) Also observe that, for ba, the AdS 3 region of (7.6) and (7.16) starts at around r≥b √ n. To summarize, the geodesics that we will study are those with θ = 0 or θ = π 2 , dθ dτ = 0 and either dϕ 1 dτ = 0 or dϕ 2 dτ = 0, respectively. The conserved momenta are restricted to L 1 =L 2 = 0, P =E < 0 while dr dτ given by the negative square root in (7.42). They will start in the asymptotic AdS 3 region, that is, they will have r ∗ >b √ n and, by construction, they will fall all the way to 156 r = 0. It is evident from (7.43) that such a particle will be traveling at a very high speed in the “Lab Frame” that is at rest at the bottom of the cap. For simplicity we will, henceforth, take n = 1. 7.4.2 Tidal forces For a geodesic with proper velocity, V μ = dx μ dτ , the equation of geodesic deviation is: A μ ≡ D 2 S μ dτ 2 = −R μ νρσ V ν S ρ V σ , (7.47) whereS ρ isthedeviationvector. Byshiftingthepropertimecoordinatesofneighboringgeodesics one can arrange S ρ V ρ = 0 over the family of geodesics. Thus S ρ is a space-like vector in the rest-frame of the geodesic observer. One can re-scale S μ at any one point so that S μ S μ = 1 and then A μ represents the acceleration per unit distance, or the tidal stress. The skew-symmetry of the Riemann tensor means that A μ V μ = 0 and so the tidal acceleration is similarly space-like, representing the tidal stress in the rest-frame of the infalling observer with velocity, V μ . To find the largest stress one can maximize the norm, q A μ A μ , of A μ over all the choices of S μ , subject to the constraint S μ S μ = 1. We will consider the geodesics defined in the previous section. To analyze the stress forces we introduce what is sometimes called the “tidal tensor:” A μ ρ ≡ −R μ νρσ V ν V σ , (7.48) and consider its norm and some of its eigenvalues and eigenvectors. In particular, we define |A| ≡ q A μ ρ A ρ μ . (7.49) Note that since V μ = dx μ dτ is dimensionless,A has the same dimensions as the curvature tensor, L −2 . 157 If V μ and S μ (a) , a = 1,... 5, are orthonormal vectors then it is trivial to see that |A| 2 = 5 X a=1 A (a) μ A (a)μ , (7.50) where A (a) μ is given by (7.47) with S μ =S (a) μ . If there is one dominant direction of maximum stress then one can adapt the basis,S (a) μ to this direction and|A| will yield this maximum stress. For our problem, the maximum stress is spread over multiple directions and so|A| will give an estimate of this stress up to a numerical factor of order 1. One should also note thatA μ ρ is not generically symmetric, and so the stress cannot always be directed along the displacement directions, S (a) μ . The scale of the stress and string transitions of probes As a warm-up exercise, we computedA μ ρ for radially infalling geodesics in the BTZ metric 3 . We obtained a result that was independent of the starting point: |A| = 2 R y b = √ 2 √ Q 1 Q 5 = √ 2 √ N 1 N 5 q Vol(T 4 ) ` 4 10 . (7.51) Asonewouldexpect,andhope,thisbecomesvanishinglysmallcomparedtothePlanck/compactification scale if N 1 and N 5 are suitably large. There is, indeed, no tidal drama as one approaches the macroscopic horizon of a black hole. The story is very different in the deep, scaling throat of the general metric, (7.6). We will assume that ba throughout. First, when an infalling geodesic is near the top of the throat, around r =b, one obtains (to leading order in b) the same result as for the BTZ black hole, (7.51). One the other hand, at r = 0 and r =a, one finds: |A| r=0 = c 1 E 2 b 2 a 4 , |A| r=a = c 2 E 2 b 2 a 4 , (7.52) 3 Remember we are taking n = 1. 158 where E is the energy 4 of the geodesic motion and (c 1 ,c 2 ) = √ 3, 2 √ 13 9 ! for θ = π 2 , (c 1 ,c 2 ) = √ 2, 3 8 s 5 2 ! for θ = 0. (7.53) If the probe is released from rest at r ∗ ∼b, then (7.46) gives |E| ∼ v u u t b R y . (7.54) (We are dropping numerical factors of order 1 throughout this discussion.) Thus we find that, for both classes of geodesic (θ = 0, π 2 ), when they arrive in the vicinity of the cap (0≤ r≤ a) the stress has a magnitude given by: |A| cap ∼ b 3 a 4 R y = b 4 a 4 1 bR y ∼ (N 1 N 5 ) 3 2 J 2 q Vol(T 4 ) ` 4 10 . (7.55) where we have used the second equation in (7.13). Note that at the bottom of the deepest scaling throats, with J = 1, this stress is super-Planckian. Such an infalling probe must become intrin- sically stringy long before it hits the cap! Indeed, the only way to avoid the stringy dissolution of the probe is if the throat is relatively shallow: J ∼ (N 1 N 5 ) β for β > 3 4 . From (7.34), this corresponds to an energy gap of E gap ∼ 1 (N 1 N 5 ) 1−β , β > 3 4 . (7.56) Tofindwherethestringtransitionmusttakeplaceinthethroat, wecomputed|A|atweighted geometric averages of a and b. Specifically we found that for r = a (1−α) b α , 0 < α < 2 3 , the dominant term controlling|A| is given by |A| throat ∼ a 2 b 2 E 2 r 6 ∼ b 2 a 2 ! 2−3α 1 bR y ∼ q Vol(T 4 ) ` 4 10 (N 1 N 5 ) 3/2 J 2 J N 1 N 5 3α . (7.57) 4 As usual with geodesics, E is the energy per unit rest mass, and so is actually dimensionless. 159 If we have the deepest possible throat, with J∼ 1, then the tidal stress hits and exceeds the Planck scale for α≤ 1/2. This implies the central result of this chapter: the probe must undergo a stringy transition as it approaches r = √ ab. It is interesting to note that the dominant stress term that we have singled out in (7.57) is proportional to a 2 r 6 . This term becomes sub-dominant for α > 2 3 and vanishes when a = 0. Indeed, for a = 0 the metric (7.6) reduces to the BTZ metric and the stress is bounded well below the Planck/compactification scale, as in (7.51). Thus the stringy transition of the probe is a feature of having the cap at the bottom of a deep throat: the stress is induced by the probe hurtling through the cap and having its course reversed by the geometry. Stress along the sphere and the “bump” The stress along the sphere directions exhibits some rather remarkable features that can be illustrated by some of the simpler eigenvectors, U ρ , and eigenvalues, λ, ofA: A μ ρ U ρ = λU ρ . (7.58) In particular, U = ∂ ∂θ is an eigenvector ofA along the entire geodesics for both θ = 0 and θ = π 2 but the eigenvalues are very different for the two classes of geodesic. For θ = 0 we find λ = a 2 r 2 b 2 4R y (r 2 +a 2 ) 2 (a 2 + b 2 2 ) 3 2 ∼ a 2 r 2 √ 2R y b (r 2 +a 2 ) 2 , (7.59) for large b. In other words, this stress remains very small, unlike the behavior expected from (7.52). For θ = 0 one can also show that U = ∂ ∂ϕ 1 is an eigenvector with the same eigenvalue, (7.59), asU = ∂ ∂θ . Atθ = 0 theϕ 1 coordinate degenerates in the same way that spherical polars degenerate at the origin ofR 2 . This observation about the U = ∂ ∂ϕ 1 being an eigenvector with the same eigenvalue as U = ∂ ∂θ reflects the fact that the eigenspace of (7.59) is two-dimensional, corresponding to the slice of tangent space described by ∂ ∂θ and ∂ ∂ϕ 1 in the neighborhood ofθ = 0. 160 We have not be able to find any other eigenvectors ofA for θ = 0 and general values of r, however at r = 0 we find that U = − R y √ 2 ∂ ∂u + R y √ 2 ∂ ∂v + ∂ ∂ϕ 2 , (7.60) is an eigenvector with eigenvalue λ = −   E 2 b 2 a 4 + 1 R y q (a 2 + b 2 2 )   , (7.61) which exhibits the growth anticipated by (7.52). The picture for θ = π 2 is very different and has a rather more interesting structure. First, U (0) = ∂ ∂θ , U (1) = ∂ ∂ϕ 1 and U (2) = ∂ ∂ϕ 2 are all eigenvectors ofA, with eigenvalues λ I , I = 0, 1, 2. One findsλ 0 =λ 2 but this is not surprising sinceϕ 2 degenerates atθ = π 2 and we are once again finding a two-dimensional eigenspace in the tangent space at θ = π 2 . The general eigenvalues are very complicated but, for ba, we have the following limits: r→∞ : λ 0 =λ 2 ∼ a 2 b 2 4 (Q 1 Q 5 ) 3 2 R 2 y r 2 , λ 1 ∼ − a 2 b 2 4 (Q 1 Q 5 ) 3 2 R 2 y r 2 , (7.62) r→a : λ 0 =λ 2 ∼ E 2 b 2 9a 4 , λ 1 ∼ − E 2 b 2 9a 4 , (7.63) r→ 0 : λ 0 =λ 2 ∼ − E 2 b 2 2a 4 , λ 1 ∼ E 2 b 2 2a 4 . (7.64) At infinity these eigenvalues fall off more rapidly than anticipated by (7.51) but, in the cap, they exhibit the same scale as indicated by (7.52). Thus, at infinity, the tidal forces along the S 3 are rapidly decoupling from the stronger tidal forces of the AdS 3 . However, in the cap the tidal forces along the sphere directions are extremely large, exhibiting the typical asymptotic behavior (7.52). Even more interesting is the leading behavior of these eigenvalues for b a. We find the 161 0.5 1.0 1.5 2.0 2.5 3.0 ρ -0.6 -0.4 -0.2 0.2 0.4 0.6 G 1 ( ρ) G 2 ( ρ) Figure 7.1: Plot of G 1 (ρ) and G 2 (ρ) showing the fluctuations of the tidal forces in the S 3 directions as the geodesic crosses the cap of the geometry. As one can see from (7.65) and (7.66), the functions G 1 (ρ) and G 2 (ρ) take values− 1 2 and + 1 2 , respectively, at ρ = 0. following leading behavior: λ 0 = E 2 b 2 a 4 G 1 (ρ), G 1 (ρ) ≡ − (2− 11ρ 2 − 18ρ 4 − 5ρ 6 + 8ρ 8 ) 4 (1 +ρ 2 )(1 +ρ 2 +ρ 4 ) 3 , (7.65) λ 1 = E 2 b 2 a 4 G 2 (ρ), G 2 (ρ) ≡ (2− 11ρ 2 − 11ρ 4 + 8ρ 6 ) 4 (1 +ρ 2 +ρ 4 ) 3 , (7.66) where ρ≡r/a. In particular G 1 and G 2 both change signs twice on the range 0<r< 2a. (See Fig. 7.1.) Recall that the microstate structure localizes around r∼ a, as is evident from the warp factor, Λ, in (7.8). It is evidently the microstate structure that introduces a very bumpy ride for the stresses in the sphere directions. 7.4.3 Redshifts and energy scales To estimate the lowest energy excitations of the CFT states that localize in various regions of the throat, we looked, once again at the dispersion relation but now at r =a (1−α) b α withba, 0<α< 2 3 and n = 1. Focussing on the long-wavelength modes that come from excitations on the S 3 and taking 162 K(r)∼ constant in the region of interest in the throat, we find ω ∼ μ a 2 b 2 ! 1−α ∼ μ J N 1 N 5 ! 1−α . (7.67) where μ is some number of order 1. The deepest throat has J = 1, and the stringy transition occurs at α = 1 2 . Thus the energy gap of states that localize near the stringy transition of the probe is: E transition ∼ 1 √ N 1 N 5 . (7.68) It is interesting to note that while the longest strands have length N 1 N 5 , in the ensemble of strands it is expected that the typical strand length peaks at k = √ N 1 N 5 and (7.68) is the energy gap for such strands. Our geodesic probe thus seems to make its stringy transition when it encounters the energy scale appropriate to the most typical strands. 7.5 Conclusions As with the older families of scaling, multi-centered microstate geometries [85,86,97,98], we have shown that the quantization of angular momentum in the new microstate geometries of [91,93,95] leads to a limit on the depth of the throat and that excitations at the bottom of this throat have a holographic energy gap that matches the maximally-twisted sector of the dual CFT. We have also examined the curvatures and the scales of the structural features of the supergravity solutions and found that they all lie well within the range of validity of the supergravity approximation. One therefore expects that the new microstate geometries represent reliable backgrounds for holographic analyses. In this chapter we used geodesics of massive particles released from rest at the top of the BTZ throat to probe the microstate geometries. In the asymptotically-flat microstate geometries these probes would represent typical infalling particles from near the would-be black hole and, as such, they have extremely high energies compared to the quanta that localize near the cap of 163 the microstate geometry. Wefoundthat,fordeepthroats,theseprobesexperienceextremely-highstressesfromgeodesic deviation and that such probes will therefore go through a stringy transition long before they get to the cap. For the deepest throats, whose states are dual to the maximally-twisted sector, the probe undergoes a stringy transition “half-way down the throat”. The energy scale associated with this transition is also the energy gap of the most typical sector of the D1-D5 system. These large stress forces come from the deviation of the microstate geometry from BTZ geometry amplified by the relativistic speed of the infalling particle. The bumps near the bottom of the microstate geometry can also play a role in the very large tidal stress, and we saw that large stress forces in the sphere directions could even change sign as the probe encountered the localized microstructure. What these probe calculations show is that deep, scaling microstate geometries only make sense in a highly constrained environment and that exposing them to the typical matter of a black-hole environment will generate highly-excited states of the system. In this sense, our geodesic probes reveal an “instability” of the BPS microstate geometries, but it is an instability that must be present as part of black-hole physics: infalling matter must scramble into excited, non-BPS microstates. 164 Chapter 8 Concluding remarks In this thesis we have studied different smooth supersymmetric solutions of classical supergrav- ity that appear in the holographic description of field theories and in the context of microstate geometry program. There are many examples of holographic field theories in which smooth so- lutions are dual to pure states of the boundary theory, whereas a singular solution has no clear holographic interpretation and represents a “thermodynamic” description. The most canonical one is renormalization group flow inN = 1 supersymmetric gauge theory which involves confine- ment and chiral symmetry breaking in the infrared [13,14,16]. The problem with this flow was to find a correct holographic description of infrared phase of the dual field theory. The wrong solution [14] was based on a singular conifold and could not capture the IR phase, but it was instrumental in finding a right one. It was demonstrated by Klebanov and Strassler that the correct solution [13] resolves the singularity by blowing up a non-trivial cycle supported by flux, and the IR phase is described by smooth and nonsingilar geometry. One can apply the same principles to construct the microstates of black hole. From the perspective of the boundary field theory microstates should be thought of as infrared phases of matter in which certain operators acquire non-trivial vev’s. The problem then becomes to find a correct gravitational dual of those states. As in the case of Klebanov-Strassler solution one can expect that the infrared phase should be described by smooth supergravity solutions with blown 165 up cycles threaded by cohomological fluxes. The main idea of the microstate program is to replace the horizons of black holes with horizonless configurations made out of branes and momentum excitations that have the same conserved charges as classical black holes. The absence of the horizon is crucial for resolution of the black hole information paradox and preserving the unitarity of quantum mechanics. The microstate geometries provide the only possible mechanism within supergravity to support the horizon-scale structure, and it uses in essential way the extra dimensions inherent in string theory combined with the non-trivial topology of spacetime [12,73]. Using a particular truncation of maximal gauged supergravity we obtained a new class of renormalization group flows and Janus solutions with the remarkable properties. From the four- dimensional perspective the flows are running to the infinite values in the parameter space and develop a singularity in the infrared. However, the uplift of the flows to M-theory is regular (except for orbifolds) and gives rise to interesting new classes of compactification geometries, with the compact manifold being a Kähler 3-fold, that are smooth in the infrared limit. Our solutions may have important applications for the holography, since they “flow up dimensions”, going from the (2 + 1)-dimensional conformal field theory on M2 branes in the UV to a (3 + 1)- dimensional field theory on intersecting M5 branes in the infrared. The appearance of extra dimension suggests the new strongly coupled phase of matter in dual ABJM theory in which charged solitons are becoming massless. Apart from the interesting holographic interpretation our flow that approaches [ AdS 5 in the infrared may be relevant to the physics of black holes. It demonstrates the same kind of mech- anism that underlies bubbled geometries used to construct microstates in five dimensions which involves the dissolution of electric M2 branes in magnetic fluxes of M5 branes through the Chern- Simons interactions. Moreover, our flow solution has the same number of supersymmetries as the three-charged black hole. At the current state of technology most of the attempts were concentrated on construction of asymptotically flat black hole microstates, but there are still no examples of microstates of asymptotically AdS 5 black holes. We hope that our flow might be 166 useful to shed a light upon how one can approach the problem of finding non-asymptotically flat microstates. Unlike the microstate story our flow involves dielectric polarization of one set of M2 branes into three sets of M5 branes that intersect on a common four-dimensional space, while in black hole physics one starts with three different sets of M2 branes which are replaced with three sets of M5 branes that intersect in a two-dimensional space. In addition, it is not obvious that dielectric polarization of supersymmetries should necessarily happen for microstates. Overall, we have two different polarizations of M2 branes into M5 branes. It would be extremely interesting to see if there is, in fact, a deeper relationship and use the results to further inform black-hole physics, or vice versa. It has been recently discovered by Martinec and Niehoff that W-branes which wrap the cycles inside microstate bubbled geometries can become light in the scaling regime and may form a condensate giving rise to a new phase of matter with exponentially growing number of BPS states. Using the brane probe analysis they argued that these states can account for the significant fraction of the black hole entropy. Raeymaekers and Van den Bleeken have proposed the back-reacted version of W-brane solution and suggested that W-branes themselves can be considered as black hole microstates. We carefully analyzed the supersymmetries of the back- reacted solution and showed that it cannot be considered as a microstate since it breaks half of supersymmetries of the black hole. Instead W-branes should rather be thought of as a one- particle excitations of the new phase of matter that appears in the scaling limit. The microstate geometries known so far have been obtained using toroidal compactifications either to five or to six dimensions. To describe a more complicated microstate structures such as W-brane condensates or broader classes of W-brane states one has to go beyond the toroidal compactification and use more general manifolds with non-trivial topology. Moreover, the fully back-reactedsolutionshouldinvolvethedynamicalfluctuationsoffieldsbothinsidethespacetime and inside the compactification manifold. We suggested an approach to supergravity description of W-brane states based on non-trivial 167 compactification of extra dimensions in which the standard T 6 is replaced by more complicated Ricci-flat manifold. This allowed us to blow up or down homology cycles without making the entire manifold singular. In particular we replaced the T 4 inside of T 6 with K3 surface and used the Gibbons-Hawking manifoldM GH to model it locally. We also promoted Kähler mod- uli of the internal Gibbons-Hawking space into dynamical fields on the spacetime. As a first step we considered purely electric fluxes and concentrated on the backgrounds that have the same amount of supersymmetry as the three-charged black hole. Our analysis of BPS equations showed that compactification manifold has to be non-trivially fibered over spacetime with the fields called compensators. The Maxwell fields also have to include compensators through frames of the fibration. Once the correct fiber structure is introduced the BPS equations can be solved automatically without any approximations! Unfortunately solving the BPS equations does not always guarantee a solution to the equations of motion. We first found a solution of Maxwell equation which is linear in moduli fields ofM GH and then extended our analysis to finite values of the moduli. It appears that the translation symmetry of the compactification manifold is broken, and so the branes do not smear out uniformly. We examined the most canonical “topo- logical solution” by adding the brane distribution on compactification manifold which is always intrinsically dipolar (with no net charge) outside δ-function sources in the space-time. It would be very interesting to generalize the above construction and find a completely smooth exact solution which is sourced entirely by non-trivial magnetic fluxes threading both the spacetime and internal manifold. We leave this work for future research. It is also extremely interesting to see what new physics at the horizon scale associated with microstate geometries can be predicted that differ them from standard black holes. To find these predictions we considered interaction between the infalling matter and microstate structure by examining tidal forces acting on the massive probe particle that moves along the radial geodesic. The microstate geometries approximate black hole metric away from the horizon but significantly differ from them at the scale of the horizon. This new horizon-scale structure must manifest itself through the interactions with the infalling matter. It is well known from General Relativity that 168 tidal force is very small for a large enough black hole since the square of the Riemann tensor is inversely proportional to the forth power of the black hole mass, and the classical observer will not experience serious disruption as one crosses the event horizon. One would imagine that capping off the black hole throat closely above the horizon will not change much the classical picture. Contrary to our expectations we found that the tidal stress reaches the Plank energy scale before the particle reaches the region where the microstate structure is localized. This means that the probe must undergo a highly energetic transition and be teared apart into its stringy constituents. We argued that such large tidal forces are caused not by the effects of large curvature, which remains bounded everywhere, but are due to the presence of microstate structure itself. It would be interesting the generalize this computation to other kinds of probes such as strings and branes and perhaps study their excitations. Our analysis suggests that ordinary matter should be trapped by microstates and somehow dissolve into a microstate structure. Previously it was thought that quantum effects, such as tunneling of matter into a vast family of degenerate states, should play a role in preventing collapsing shell from forming a horizon. We have found that there is purely classical, tidal phenomenon that leads to stringy transition that will ultimately scramble the probe. Trapping of particles also implies that part of the energy of the infalling matter should be absorbed by microstates, and this should create some mild supersymmetry breaking excitations of microstates. It would be extremely interesting to take into account the back-reaction of the nearby matter on microstate geometries and construct these excitations. It is also expected that the non-supersymmetric excitations will eventually be emitted in some form of Hawking radiation. Exploring the details of this process should answer the question how microstate data is encoded in that radiation, and how one can restore the information about the initial state of matter that formed the microstate. Eventually, this should provide a constructive resolution of the Hawking paradox. The ultimate goal of these studies is to reveal the observable signatures of new physics at the horizon scale associated with microstructures that will allow us to distinguish them from black holes and in so doing enable encoding and extraction of black hole information. 169 Appendices 170 Appendix A Conventions We use the same conventions as in [69] with the “mostly plus metric” and the eleven-dimensional equations of motion given by R MN +g MN R = 1 3 F MPQR F N PQR , (A.1) ∇ M F MNPQ = − 1 576 1 √ −g NPQR 1 ...R 8 F R 1 ...R 4 F R 5 ...R 8 . (A.2) The Maxwell equation can be rewritten in terms of forms as 1 d?F (4) +F (4) ∧F (4) = 0, (A.3) where ?≡∗ 1,10 is the Hodge dual in eleven dimensions. In general, we define the Hodge dual of a k-form, ω, in d-dimensions by (∗ω) i 1 ...i d−k = 1 k! η i 1 ...i d−k j 1 ...j k ω j 1 ...j k , (A.4) where η i 1 ...i d ≡ 1 q |g| i 1 ...i d , i 1 ...i d = 1. (A.5) 1 Note that our normalization of the four-form flux is according to the “old supergravity convention.” 171 Then (∗ω)∧ω = ±|ω| 2 vol, |ω| 2 ≡ 1 k! ω i 1 ...i k ω i 1 ...i k . (A.6) with the + sign is for a positive definite metric and the− sign for a Minkowski signature mostly plus metric. For a (p,q)-form Ω (p,q) onM 1,3 ×M 7 with the warped product metric (3.43), we have a convenient decomposition of the Hodge dual: ? Ω (p,q) = (−1) p(7−q) ∗ 1,3 ∗ 7 Ω (p,q) , (A.7) where∗ 1,3 and∗ 7 are, respectively, the dual onM 1,3 with respect to the four-dimensional part of the metric, g μν , and the dual onM 7 with the internal metric, g mn . Factoring out the warp factor, we have ∗ 1,3 ω (p) = Δ p−2 ◦ ∗ 1,3 ω (p) , (A.8) where ω (p) is a p-form onM 1,3 and ◦ ∗ 1,3 is the dual with respect to ◦ g μν . 172 Appendix B Reduced E 7(7) tensors and the scalar action Using the 56-bein (3.4) and with the SO(8) gauge field set to zero, the SU(8) composite gauge field of theN =8 theory, A μ ijkl ≡ A μ [ijkl] = − 2 √ 2 u ij IJ ∂ μ v klIJ −v ijIJ ∂ μ u kl IJ , (B.1) has the following non-vanishing components: A μ 1234 = A μ 1256 = A μ 1278 = A μ 3456 = A μ 3478 = A μ 5678 = − √ 2∂ μ ¯ z 1−|z| 2 . (B.2) Hence the kinetic action of the scalar fields is e −1 L kin ≡ − 1 96 A μijkl A μ ijkl = − 3 ∂ μ z∂ μ ¯ z (1−|z| 2 ) 2 = − 3∂ μ λ∂ μ λ− 3 4 sinh 2 (2λ)∂ μ ζ∂ μ ζ. (B.3) 173 Similarly, for the A-tensors, A ij 1 ≡A ji 1 and A 2i jkl ≡A 2i [jkl] , we have: A 11 1 = ... = A 66 1 = 1 +z¯ z 2 (1−|z| 2 ) 3/2 , A 77 1 = A 88 1 = 1 +z 3 (1−|z| 2 ) 3/2 , (B.4) and A 2 1 234 = A 2 1 256 = A 2 3 124 = A 2 3 456 = A 2 5 126 = A 2 5 346 = − (1 +z)¯ z (1−|z| 2 ) 3/2 , A 2 2 134 = A 2 2 156 = A 2 4 123 = A 2 4 356 = A 2 6 125 = A 2 6 345 = (1 +z)¯ z (1−|z| 2 ) 3/2 , A 2 1 278 = A 2 3 478 = A 2 5 678 = − (1 +z)z (1−|z| 2 ) 3/2 , A 2 2 178 = A 2 4 378 = A 2 6 578 = (1 +z)z (1−|z| 2 ) 3/2 , A 2 7 128 = A 2 7 348 = A 2 7 568 = − z + ¯ z 2 (1−|z| 2 ) 3/2 , A 2 8 127 = A 2 8 347 = A 2 8 567 = z + ¯ z 2 (1−|z| 2 ) 3/2 . (B.5) Then the scalar potential is P ≡ − 3 4 A 1 ij 2 − 1 24 A 2i jkl 2 = − 6(1 +|z| 2 ) 1−|z| 2 = − 6 cosh(2λ). (B.6) 174 Appendix C The Freund-Rubin flux Thecalculationofthespace-timepartoftheflux,F st (4) , usingthemethodemployedinsection3.3.5 is quite involved even for much simpler solutions such as uplifts of stationary points. For the latter solutions only the Freund-Rubin part of the space-time flux is present, so that F st (4) =f FR v ◦ ol 1,3 , (C.1) isproportionaltothevolumeofthefour-dimensionalspace-time,M 1,3 , wheretheproportionality constant is determined universally by the scalar potential of the four-dimensional theory [67]. This has been generalized recently in [60] to uplifts of arbitrary solutions by including corrections proportional to derivatives of the scalar potential. The new conjectured formula for the Freund- Rubin flux,f FR , reads f FR = m 7 2 P− 1 24 Q ijklb Σ ijkl +h.c. . (C.2) The Q ijkl tensor is proportional to the first variation of the potential,P, along the noncompact generators of E 7(7) acting on the scalar coset, E 7(7) /SU(8). It is given by [?] Q ijkl = 3 4 A 2m n[ij A kl]m 2n −A 1 m[i A 2m jkl] . (C.3) 175 The second tensor in (C.2) is a self-dual contraction b Σ ijkl = (u ij IJ u kl KL −v ijIJ v klKL )K IJKL , (C.4) where K IJKL = ◦ g mn K [IJ m K KL] n . (C.5) Note that at a stationary point of the scalar potential, theQ-tensor becomes anti-self-dual [?] and hence the contraction terms in (C.2) vanish. Specializing the contraction in (C.2) to the present solution we find Q ijklb Σ ijkl +h.c. = − 16ξ sinh(2λ) cosζ. (C.6) Then, using (3.13), (3.62) and (3.91), we obtain f FR = m 7 2 − 6 cosh(2λ) + 2(1− 4 sin 2 χ) sinh(2λ) cosζ = m 7 3 U, (C.7) which agrees with the calculation of the space-time flux in section 3.3.5. 176 Appendix D The Ricci tensor The non-vanishing coefficients of the diagonal components of the Ricci tensor, R MM , as defined in (3.113): A 1 = 1 6X 13/3 Σ 8/3 Ξ 2 p 2 + 1 ΣX p 2 +X 2 + 1 − 8 p 2 + 1 2 X 2 + Σ 2 3 p 3 +p 2 + 3p 2 X 4 − 2 3p 4 + 7p 2 + 4 X 2 B 1 = X 2 6X 13/3 Σ 8/3 Ξ 2 − 8p 2 X 2 + 16p 2 ΣX + Σ 2 3p 4 − 2p 2 3X 2 + 1 + 3 X 2 − 1 2 , C 1 = pX 3X 13/3 Σ 8/3 Ξ 2 − 4ΣX 3p 2 +X 2 + 3 + 8 p 2 + 1 X 2 + Σ 2 −3p 4 +p 2 6X 2 − 2 − 3X 4 + 10X 2 + 1 , D 1 = 2m −2 7 3X 4/3 Σ 8/3 Σ 2 6X 2 −g 2 m 2 7 3p 2 + 3X 2 + 4 − 2ΣX g 2 m 2 7 + 2p 2 + 2 − 2 p 2 + 1 X 2 ; (D.1) 177 A 4 = 1 3X 13/3 Σ 8/3 Ξ 2 2 p 2 + 1 2 3p 2 − 1 X 2 − 2 3p 4 + 2p 2 − 1 ΣX p 2 +X 2 + 1 + Σ 2 3 p 3 +p 2 + 3p 2 X 4 − 2 p 2 + 1 X 2 , B 4 = 1 3X 7/3 Σ 8/3 Ξ 2 − 2p 2 ΣX 3p 2 + 3X 2 − 5 + 2 3p 2 − 1 p 2 X 2 + Σ 2 3p 4 − 2p 2 + 3 X 2 − 1 2 C 4 = 1 3X 10/3 Σ 8/3 Ξ 2 2ΣX 3p 2 p 2 +X 2 +X 2 − 3 − 2 3p 4 + 2p 2 − 1 X 2 + Σ 2 −3p 4 − 2p 2 − 3X 4 + 4X 2 + 1 , D 4 = m −2 7 3X 4/3 Σ 8/3 2Σ 2 −3m 2 7 g 2 p 2 +X 2 − 4m 2 7 g 2 + 6X 2 − 4ΣX m 2 7 g 2 + 2p 2 + 2 − 4 p 2 + 1 X 2 ; (D.2) A 5 = − 4 3X 10/3 Σ 8/3 Ξ 2 p 2 + 1 (Σ−X) p 2 − ΣX + 1 , B 5 = 4p 2 X(Σ−X) 2 3X 10/3 Σ 8/3 Ξ 2 , C 5 = − 4p 3X 10/3 Σ 8/3 Ξ 2 (Σ−X) Σ p 2 +X 2 + 1 − 2 p 2 + 1 X , D 5 = 2 3Σ 8/3 X 4/3 − 2ΣX g 2 − 4m 2 7 p 2 + 1 + 2g 2 Σ 2 +m 2 7 4p 2 + 1 X 2 ; (D.3) 178 A 6 = 2 3X 10/3 Σ 8/3 Ξ 2 p 2 + 1 (Σ−X) p 2 − ΣX + 1 , B 6 = − 2p 2 (Σ−X) 2 3Σ 8/3 X 7/3 Ξ 2 , C 6 = 2p 3X 10/3 Σ 8/3 Ξ 2 (Σ−X) Σ p 2 +X 2 + 1 − 2 p 2 + 1 X , D 6 = 2 3X 4/3 Σ 8/3 Σ 2 m 2 7 9p 2 + 6 −g 2 + ΣX g 2 + 2m 2 7 p 2 + 1 +m 2 7 p 2 + 1 X 2 ; (D.4) A 10 = 2 3X 10/3 Σ 8/3 Ξ 2 3p 4 + 4p 2 + 1 (Σ−X) p 2 − ΣX + 1 , B 10 = 2p 2 X 3X 10/3 Σ 8/3 Ξ 2 (Σ−X) h 3p 2 + 1 X + Σ 2− 3X 2 i , C 10 = − 4p 3X 10/3 Σ 8/3 Ξ 2 (Σ−X) Σ − 3p 2 + 2 X 2 +p 2 + 1 + 3p 4 + 4p 2 + 1 X , D 10 = − 2 3X 4/3 Σ 8/3 2ΣX g 2 − 4m 2 7 p 2 + 1 − 2g 2 Σ 2 −m 2 7 4p 2 + 1 X 2 ; (D.5) 179 A 11 = 2 3X 13/3 Σ 8/3 Ξ 2 p 2 + 1 2 3p 2 − 1 X 2 − 3p 4 + 2p 2 − 1 ΣX p 2 +X 2 + 1 + Σ 2 2 3 p 3 +p 2 + 3p 2 X 4 − 2 p 2 + 1 X 2 , B 11 = 1 3X 7/3 Σ 8/3 Ξ 2 2 3p 2 − 1 p 2 X 2 − 2p 2 ΣX 3p 2 + 3X 2 − 5 + Σ 2 3p 4 − 2p 2 + 3 X 2 − 1 2 , C 11 = 2p 3X 10/3 Σ 8/3 Ξ 2 2ΣX 3p 2 p 2 +X 2 +X 2 − 3 − 2 3p 4 + 2p 2 − 1 X 2 + Σ 2 −3p 4 − 2p 2 − 3X 4 + 4X 2 + 1 , D 11 = 2 3X 4/3 Σ 8/3 − Σ 2 3g 2 p 2 −g 2 3X 2 + 1 + 6m 2 7 X 2 + 1 − 4ΣX m 2 7 p 2 + 1 −g 2 −m 2 7 2p 2 − 1 X 2 . 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Proceedings, Theoretical Advanced Study Institute, TASI’99, Boulder, USA, May 31-June 25, 1999, pages 353–433, 2000. 190 List of Figures 2.1 The contour map ofV (on the left) andW (on the right), withϕ 1 on the vertical axis and α = 1 √ 6 ϕ 3 on the horizontal axis. The five labeled points are the only extrema ofV in this plane. A numerical solution of the steepest descent equations is shown superimposed on the contour plot of W [33]. . . . . . . . . . . . . . . . . 19 3.1 TypicalflowtrajectoriesfortheJanussolutionstotheBPSequations(3.19)–(3.21) in the (λ cosζ,λ sinζ)-plane. The background contours are of the superpotential W (λ,ζ). A red dot denotes the “central point” of a flow at (λ c cosζ c ,λ c sinζ c ) where A 0 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Typical profiles of the metric function, A(r), and the scalar fields, λ(r) and ζ(r), for the different types of flows in Figure 3.1. . . . . . . . . . . . . . . . . . . . . . 30 3.3 RG-flow trajectories in the (λ cosζ,λ sinζ)-plane. The background contours are of the real superpotential W (λ,ζ). The ridge trajectories have constant ζ with cos 3ζ = 1 (green) and cos 3ζ =−1 (red), respectively. . . . . . . . . . . . . . . . 31 3.4 Ridge flows for cos 3ζ =−1 (red) and cos 3ζ = 1 (green) with A 0 = 0. . . . . . . . 32 4.1 This figure depicts some non-trivial cycles of the Gibbons-Hawking geometry. The behaviour of the U(1) fiber is shown along curves between the sources of the potential,V. Here the fibers sweep out a pair of intersecting homology spheres [73]. 81 191 4.2 Geometric transition of black ring: The first diagram shows the geometry before the transition. The second shows the resolved geometry, where a pair of GH charges has nucleated at positions a and b. . . . . . . . . . . . . . . . . . . . . . 86 7.1 Plot ofG 1 (ρ) andG 2 (ρ) showing the fluctuations of the tidal forces in theS 3 direc- tions as the geodesic crosses the cap of the geometry. As one can see from (7.65) and (7.66), the functions G 1 (ρ) and G 2 (ρ) take values− 1 2 and + 1 2 , respectively, at ρ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 192 List of Tables 4.1 Layout of the branes that give the supertubes and black rings in an M-theory duality frame. Vertical arrowsl, indicate the directions along which the branes are extended, and horizontal arrows,↔, indicate the smearing directions. The functions, y μ (σ), indicate that the brane wraps a simple closed curve inR 4 that defines the black-ring or supertube profile. A star, ?, indicates that a brane is smeared along the supertube profile, and pointlike on the other three directions [73]. 77 193

Asset Metadata

Creator Tyukov, Alexander Vasilevich (author)

Core Title Supersymmetric solutions and black hole microstate geometries

Contributor Electronically uploaded by the author (provenance)

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program physics

Publication Date 07/26/2018

Defense Date 05/01/2018

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

Tag black hole,holography,microstate,OAI-PMH Harvest,renormalization group flow,supergravity,supersymmetry

Format application/pdf (imt)

Language English

Advisor Warner, Nicholas (committee chair), Haas, Stephan (committee member), Lidar, Daniel (committee member), Nemeschansky, Dennis (committee member), Pilch, Krzysztof (committee member)

Creator Email alex_t06@mail.ru,tyukov.alexander@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-23602

Unique identifier UC11668803

Identifier etd-TyukovAlex-6488.pdf (filename),usctheses-c89-23602 (legacy record id)

Legacy Identifier etd-TyukovAlex-6488.pdf

Dmrecord 23602

Document Type Dissertation

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Rights Tyukov, Alexander Vasilevich

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

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA

Abstract (if available)

Abstract In this thesis we investigate different aspects of supersymmetric microstate geometries and construct the renormalization group flows in holographic field theory. The idea of the microstate geometry program is to use the string theory as a consistent theory of quantum gravity to constructively resolve the long-standing black hole information paradox and the related entropy problem by replacing the classical black hole with the horizonless geometry. The microstate geometries are smooth, horizonless solutions of classical supergravity equations which have the same conserved charges as black hole and allow for unitary scattering. There is a hope that one can find enough of them to account for the huge gravitational entropy of black hole at least at the semi-classical level. The construction of such microstates crucially relies on the presence of extra-dimensions, the non-trivial topology of spacetime and the long-ranged supergravity fluxes. ❧ In the first part we consider the supersymmetric renormalization group flows in gauged supergravity which share some of the properties of the black hole microstate geometries such as presence of the large topological cycles threaded by purely magnetic fluxes. The infra-red limit of the solutions appears to be singular from four-dimensional perspective