Supersymmetric Partition Functions in the Ads/CFT Conjecture
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Supersymmetric Partition Functions in the AdS/CFT Conjecture A dissertation presented by Suvrat Raju to The Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Physics Harvard University Cambridge, Massachusetts June 2008 CC BY: $n C This work is licenced under the Creative Commons Attribution-Noncommercial-Share Alike 3.0 License. You may freely copy, distribute, transmit or adapt this work under the following conditions: $n You may not use this work for commercial purposes. C If you alter, transform, or build upon this work, you may distribute the resulting work only under the same or similar license to this one. BY: You must attribute this work. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, CA 94105, USA. Thesis advisor Author Shiraz Minwalla Suvrat Raju Supersymmetric Partition Functions in the AdS/CFT Conjec- ture Abstract We study supersymmetric partition functions in several versions of the AdS/CFT correspondence. We present an Index for superconformal field theories in d = 3; 4; 5; 6. This captures all information about the spectrum that is protected, under continuous de- formations of the theory, purely by group theory. We compute our Index in N = 4 SYM at weak coupling using gauge theory and at strong coupling using supergravity and find perfect agreement at large N. We also compute this Index for supergravity 7 4 on AdS4 × S and AdS7 × S and for the recently constructed Chern Simons matter theories. We count 1/16 BPS states in the free gauge theory and find qualitative agreement with the entropy of big black holes in AdS5. We note that the near horizon geometry of some small supersymmetric black holes is an extremal BTZ black holes fibered on a compact base and propose a possible explanation for this, based on giant gravitons. We also find the partition function of the chiral ring of the N = 4 SYM theory at finite coupling and finite N. Turning to AdS3, we study the low energy 1/4 and 1/2 BPS partition functions by finding all classical supersymmetric probe brane solutions of string theory on this iii Abstract iv 3 4 background. If the background BNS field and theta angle vanish, AdS3 ×S ×T =K3 supports supersymmetric probes: D1 branes, D5 branes and bound states of D5 and D1 branes. In global AdS, upon quantization, these solutions give rise to states in discrete representations of the SL(2,R) WZW model on AdS3. We conclude that (a) the 1/4 BPS partition function jumps if we turn on a theta angle or NS-NS field (b) generic 1/2 BPS states are protected. We successfully com- pare our 1/2 BPS partition function with that of the symmetric product. We also discuss puzzles, and their possible resolutions, in reproducing the elliptic genus of the symmetric product. Finally, we comment on the spectrum of particles in the theory of gravity dual to non-supersymmetric Yang Mills theory on S3× time. Contents Previously Published Work . vii Acknowledgments . viii 1 Introduction 1 1.1 Background . .1 1.2 Motivation . .4 1.3 Summary of Results . .5 2 An Index for 4 Dimensional Superconformal Field Theories 10 2.1 Introduction . 10 2.2 Unitary Representations of 4 dimensional Superconformal Algebras . 13 2.3 A Trace Formula for the Index . 25 2.4 Computation of the Index in N = 4 Yang Mills on S3 ......... 31 2.5 Discussion . 39 3 Supersymmetric Partition Functions in AdS5/CFT4 41 3.1 Introduction . 41 1 3.2 The partition function over 16 BPS states . 43 3.3 Small Supersymmetric Black Holes in AdS5 as Giant Gravitons . 53 3.4 Near Horizon Geometry of Supersymmetric Black Holes in AdS5 ... 56 1 1 th 1 th 3.5 Partition Functions over 2 , 4 and 8 BPS States . 66 3.6 Comparing the Cohomological Partition Function and the Index . 74 3.7 Discussion . 77 4 Indices for Superconformal Field Theories in 3; 5 and 6 Dimensions 79 4.1 Introduction . 79 4.2 d=3 . 81 4.3 d=6 . 105 4.4 d=5 . 127 4.5 Discussion . 135 v Contents vi 5 Supersymmetric States in AdS3/CFT2 I : Classical Analysis 137 5.1 Introduction . 137 5.2 Killing spinor and kappa symmetry analysis . 141 5.3 Charge Analysis: D strings . 159 5.4 Charge Analysis: D1-D5 bound state probes . 172 5.5 Moving off the Special Point in Moduli Space . 184 5.6 Semi-Classical Quantization . 191 5.7 Results and Discussion . 197 5.A Miscellaneous Technical Details . 199 6 Supersymmetric States in AdS3/CFT2 II : Quantum Analysis 202 6.1 Introduction . 202 6.2 Classical Supersymmetric Solutions in Global AdS3 .......... 208 6.3 Another Approach to Classical Solutions: `Polyakov' Action . 218 6.4 Semi-Classical Quantization . 230 6.5 Exact Analysis of the D-string . 246 6.6 Results . 258 6.A Technical Details of the Spacetime Partition Function . 260 6.B Theta Functions . 267 7 Future Directions 268 A Moving Away from Supersymmetry 270 A.1 Introduction . 270 A.2 Conformal Algebra . 271 A.3 Representations of the Conformal Group . 274 A.4 Characters . 276 A.5 Oscillator Construction . 278 A.6 Character Decomposition . 279 A.7 The Integral . 282 A.8 Proof of Integral Multiplicities . 288 A.9 N =4SYM................................ 290 A.10 Summary . 291 Bibliography 292 Previously Published Work 1. Chapters 1 and 2 of this thesis are based largely on the paper \An Index for 4 dimensional super conformal theories" J. Kinney, J. M. Maldacena, S. Minwalla and S. Raju Commun. Math. Phys. 275, 209 (2007) [arXiv:hep-th/0510251] 2. In addition, Chapter 2 also includes material from the unpublished work \Near Horizon Geometry of Extremal Black Holes in AdS5" S. Kim, S. Minwalla, S. Raju, S. Trivedi 3. Chapter 3 is based on the paper \Indices for Superconformal Field Theories in 3,5 and 6 Dimensions" J. Bhattacharya, S. Bhattacharyya, S. Minwalla and S. Raju JHEP 0802, 064 (2008) [arXiv:0801.1435 [hep-th]] 4. Chapter 4 is based on the paper \Supersymmetric Giant Graviton Solutions in AdS3" G. Mandal, S. Raju and M. Smedback Phys. Rev. D 77, 046011 (2008) [arXiv:0709.1168 [hep-th]] 5. Chapter 5 is based on the paper \Counting Giant Gravitons in AdS3" S. Raju Phys. Rev. D 77, 046012 (2008) [arXiv:0709.1171 [hep-th]] 6. Appendix A is based on the paper \The spectrum of Yang Mills on a sphere" A. Barabanschikov, L. Grant, L. L. Huang and S. Raju JHEP 0601, 160 (2006) [arXiv:hep-th/0501063] vii Acknowledgments First, I would like to thank my advisor Shiraz Minwalla for his advice and encour- agement throughout my PhD work. The physics department at Harvard provided a very vibrant intellectual atmo- sphere for my studies. I benefited tremendously from interactions with my fellow graduate students and from many discussions about physics, philosophy, politics and everything else in the universe and beyond it! I would especially like to thank my many friends and colleagues in the physics department { Lars Grant, Kyriakos Papadodi- mas, Joe Marsano, Subhaneil Lahiri, Monica Guica, Mohammad Hafezi, Philippe Larochelle, Morten Ernebjerg, Itay Yavin, Josh Lapan, Kirill Saraikin, Xi Yin, Geor- gios Pastras, Jesse Thaler, Mboyo Esole, Clay Cordova, David Simmons-Duffin, Ma- son Klein, Yina Mo, Christine Wang, Param Dhillon, Wei Li, Damon Clark, Michael Hohensee, Daniel Babich and John Paul Chou. I would like to thank Jihye Seo for comments on a draft on this thesis. Lubos Motl and Nima Arkani-Hamed and later Frederik Denef ensured that the fourth and fifth floors of Jefferson were always lively and exciting. I would like to thank Peter Galison for introducing me to the History of Science group at Harvard and for several educative discussions. Although my interaction with Sidney Coleman was limited, my understanding of quantum field theory is largely due to him. I would like to acknowledge the administrative staff at Harvard, particularly Sheila Ferguson and Dayle Maynard, for their patient help with everything. I spent part of my PhD at TIFR and I would like to thank Rudra Jena, Rahul Nigam, Suresh Nampuri, R. Loganayagam, Anindya Mukherjee, Pallab Basu, Ashiq Iqbal, Argha Banerjee, Sayantani Bhattacharyya, Jyotirmoy Bhattacharya, Apoorva viii Acknowledgments ix Nagar and everyone else in the theory student room. I would also like to thank K. Narayan, Kevin Goldstein, Sandip Trivedi and Gautam Mandal for many discussions. I would like to acknowledge my many friends outside the the physics department, particularly Dilini Pinnaduwage, Antara Datta, Kumar Sambhav and Shruti Mo- hanty. Manasi was a source of constant support and even made the Mumbai local trains bearable. An extremely important influence on me, in my years as a graduate student, came from my fellow activists. I would like to acknowledge everyone in Boston, who worked with me as part of the Harvard Initiative for Peace and Justice, the Socialist Alternative, the Boston Stop the Wars Coalition, the Association for India's Development, the Alliance for a Secular and Democratic South Asia and particularly Kaveri who was a constant comrade. In India I would like to acknowledge the members of the Young Activist Move- ment, the Lokshahi Hakka Sangathana, the Research Unit in Political Economy, the National Alliance for Peoples Movements in Mumbai and the TIFR Education Group. I was shaped by my family and to them I dedicate this thesis. To Ma, Pa and Auchu. x Chapter 1 Introduction 1.1 Background Physical theories are judged by both internal and external consistency.