Internationl School for Advanced Studies (SISSA/Trieste) Pure spinor formalism and gauge/string duality by Houman Safaai Supervisor Giulio Bonelli A thesis submitted in partial ful¯llment for the degree of Doctor of Philosophy in the Sector of Elementary Particles Scuola Internazionale Superiore di Studi Avanzati September 2009 \ And that inverted Bowl we call The Sky, Whereunder crawling coop't we live and die, Lift not your hands to It for help¡for It. As impotently moves as you or I." Omar Khayym (11th century), Persian astronomer and poet, Rubiyt of Omar Khayym Abstract Sector of Elementary Particles Scuola Internazionale Superiore di Studi Avanzati Doctor of Philosophy by Houman Safaai A worldsheet interpretation of AdS/CFT duality from the point of view of a topological open/closed string duality using the power of pure spinor formalism will be studied. We will show that the pure spinor superstring on some maximally supersymmetric back- grounds which admit a particular Z4 automorphism can be recasted as a topological A-model action on a fermionic coset plus a BRST exact term. This topological model will be interpreted as the superstring theory at zero radius. Using this decomposition we will prove the exactness of the σ-model on these backgrounds. We then show that corresponding to this topological model, there exist a gauged linear sigma model which makes it possible to sketch the superstring theory in the small radius limit as the dual limit of the perturbative gauge theory. Studying the branch geometry of this gauged sigma model in di®erent phases gives information about how the gauge/string duality is realized at small radius from a similar point of view of the topological open/closed conifold duality studied by Gopakumar, Ooguri and Vafa. Moreover, we will discuss possible D-brane boundary conditions in this model. Using this D-branes, we will make 5 an exact check in the N = 4 SYM/AdS5 £ S duality. We will show that the exact computation of the expectation value of the circular Wilson loops in the gauge theory side can be obtained from the amplitudes of some particular D-branes as the dual of the Wilson loops in the superstring side. The next step will be to construct a BV action for G=G principal chiral model with G 2 P SU(2; 2j4), we will show that after applying di®erent gauge ¯xings of the model, we will get either a topological A-model theory or a topological thery whose supersymmetric charge is equal to the pure spinor BRST charge. Using this model one can explore the cohomology of the pure spinor action from the topological BV model. Then we show that there exist a particular consistent defor- mation of the G=G action equal to the pure spinor superstring action. In this way we generate the superstring action on a non-zero radius AdS background as a perturbation around a topological model corresponding to zero radius limit of the superstring theory. Using this picture we will give an argument based on the worldsheet interpretation of open/closed duality to give a worldsheet explanation for AdS/CFT duality. A better understanding of this picture might give a prove of Maldacena's conjecture. At the end we will show that using the topological A-model, one can also give a prescription for 5 computing multiloop amplitudes in the superstring theory on AdS5 £ S background. Acknowledgements Tracing back my interest into physics reminds me the old days of the early-school time where I, together with very ambitious friends, was always dreaming about physics and deeply indulgated by the beauty of science. I would like to thank all those friends and teachers and very specially my parrents for all their support, encouragement and love they gave me during my life to help me pursuing my dreams. Anything I get in my scienti¯c carrier is the crop of the seeds they planted years ago by their immaculate training and never-ending sacri¯ces. I would like to express my deep thanks to my supervisor in SISSA, Dr. Giulio Bonelli, for all the suuport, time and scienti¯c knowledge he gave me. I enjoyed his way of thinking and working in which I learned many things from him. We spent many hours of our lifes together discussing about physics but I should admit that the part out of the work and the fact that I could count on him as a near and supportive frined was invaluable for me too. I bene¯ted a lot in the collaborations and discussions with Dr. Pietro Antonio Grassi to whom I want to express my thanks for all the things I learned from him and all his support. It is a pleasure for me also to thank my external referee, Prof. Dimitry Sorokin, for his valuable comments and suggestions on the thesis. I would like to thank also my ex- supervisor in Iran, Prof. H. Arfaei, for many things he learned me and for his guidance in the early stages of my scinti¯c carrier. Also I am thankful to all my professors in SISSA. I enjoyed sharing many deep and happy moments of my life with many good friends in Trieste during these four years. Their presence made the life here more familiar and melted the ices of the lonliness. I am thankful to all of them specially to my very early and endless friends Andrea Brini, Davide Forcella and Diego Gallego who has been very near during all these years. Last but of course not least, I would like to thank deeply Ladan, who has been beside me as the nearest one during the last stages of my PhD, for all the invaluable friendship, support, extreme happiness, energy and love she gave me. Trieste, September 2009 v Contents Abstract iv Acknowledgements v 1 Introduction 1 1.1 General gauge/string duality ......................... 1 1.2 AdS/CFT correspondence ........................... 5 1.3 Topological A-model conifold open/closed duality .............. 9 1.4 Towards a worldsheet proof of AdS/CFT duality .............. 17 Conformal exactness of the background: ........... 24 An exact check of AdS/CFT duality: ............. 24 Multiloop amplitudes computation: .............. 26 1.5 G=G principal chiral model and its deformation ............... 26 1.6 R¶esum¶eof the thesis .............................. 29 2 Pure spinor formalism of superstring theory 33 2.1 Green-Schwartz formalism of superstrings .................. 35 5 3 2.1.1 Structure of AdS5 £ S and AdS~ 4 £ CP supercosets ........ 38 2.1.2 Sigma model action for supercosets with Z4 automorphism .... 41 2.2 Pure spinor action for flat background .................... 45 2.3 Pure spinor formalism for curved backgrounds ................ 50 2.3.1 Pure spinor action for superstring on AdS5 £ S5 .......... 52 3 2.3.2 Pure spinor action for superstring on AdS~ 4 £ CP ......... 56 3 Topological decomposition of pure spinor superstring action 63 3.1 Topological A-model on a Grassmannian ................... 63 3.1.1 From KÄahleraction to topological A-model action ......... 67 3.2 Pure spinor superstring on supercosets with Z4 automorphism and the \bonus\ symmetry ............................... 73 3.3 Mapping pure spinor superstring action to topological A-model action .. 77 3.4 On conformal exactness of superstring backgrounds ............. 81 4 An exact check of AdS/CFT duality using the topological A-model 83 4.1 Gauged linear sigma-model for the superstring action ............ 83 5 4.1.1 Gauged linear sigma-model of AdS5 £ S background ....... 84 vii viii 3 4.1.2 Gauged linear sigma-model of AdS~ 4 £ CP background ...... 85 4.2 Vacua of the gauged linear sigma-model ................... 87 4.3 Open sector and D-branes ........................... 91 5 4.3.1 Open sector of topological AdS5 £ S ................ 91 3 4.3.2 Open sector of the topological AdS~ 4 £ CP ............. 92 5 4.4 An exact check in AdS5 £ S =N = 4; d = 4 SYM duality .......... 94 4.4.1 Mirror symmetry, superconifold and matrix model ......... 94 4.4.1.1 Mirror symmetry ....................... 94 First description: ........................ 96 Second description: ....................... 97 4.4.1.2 Geometric transition ..................... 105 4.4.1.3 From open topological B-model superstring to holomor- phic Chern-Simons ...................... 106 4.4.1.4 Dimensional reduction and The Gaussian Matrix model . 109 4.4.2 Circular Wilson loop and its dual in topological model ....... 111 4.4.2.1 Circular Wilson loops in N = 4 d = 4 SYM ........ 112 4.4.2.2 Dual of the circular Wilson loops in the superstring ... 115 5 The Anti¯eld Lagrangian quantization of gauge theories 119 5.1 Basics of gauge theories ............................ 119 5.2 Faddeev-Popov quantization procedure .................... 123 5.3 BRST quantization of gauge theories ..................... 127 5.4 The BV anti¯eld formalism .......................... 130 5.5 Consistent deformation of the BV action ................... 138 6 Towards a worldsheet description of AdS/CFT duality 141 6.1 BV action for the G=G principal chiral model ................ 142 6.2 Gauge-¯xing the G=G principal chiral model ................ 147 6.2.1 Gauge ¯xing to topological A-model ................. 147 6.2.2 Gauge ¯xing to a topological model with Qtop = Qpure spinor .... 151 6.3 Consistent deformation of the G=G model .................. 158 6.3.1 Descent equations ........................... 161 6.4 A worldsheet description of AdS/CFT duality ................ 164 7 Amplitudes computation 167 5 7.1 Pure spinor amplitude for AdS5 £ S ..................... 167 7.2 Topological A-model amplitude ........................ 169 5 7.3 Topological A-model of AdS5 £ S ...................... 172 The case g = 0: ......................... 174 The case g = 1: ......................... 175 The case g > 1: ......................... 176 8 Open questions and outlook 179 To my parents, Parvaneh and Behrouz And to the one I love.