Field Theory On Cosmological Spacetime: Some Results from AdS/CFT By Soumyabrata Chatterjee PHYS07201104003 Institute Of Physics, Bhubaneswar A thesis submitted to the Board of Studies in Physical Sciences In partial fulfillment of requirements For the Degree of DOCTOR OF PHILOSOPHY of HOMI BHABHA NATIONAL INSTITUTE December, 2017 2 Homi Bhabha National Institute Recommendations of the Viva Voce Board As members of the Viva Voce Board, we certify that we have read the dissertation pre- pared by Soumyabrata Chatterjee entitled “Field theory on cosmological spacetime: some results from AdS/CFT” and recommend that it maybe accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy. Date: Chair - Prof. Arun M. Jayannavar Date: Guide - Prof. Sudipta Mukherji Date: External examiner - Date: Member 1 - Dr. Amitabh Virmani Date: Member 2 - Dr. Sanjib K. Agarwalla Final approval and acceptance of this dissertation is contingent upon the candidate’s submission of the final copies of the dissertation to HBNI. I hereby certify that I have read this dissertation prepared under my direction and recommend that it may be accepted as fulfilling the dissertation requirement. Date: Place: (Prof. Sudipta Mukherji) STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfilment of requirements for an advanced degree at Homi Bhabha National Institute (HBNI) and is deposited in the Library to be made available to borrowers under rules of the HBNI. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgement of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the Competent Authority of HBNI when in his or her judgement the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. Soumyabrata Chatterjee DECLARATION I, hereby declare that the investigation presented in the thesis has been carried out by me. The work is original and has not been submitted earlier as a whole or in part for a degree / diploma at this or any other Institution / University. Soumyabrata Chatterjee To My Family 6 ACKNOWLEDGEMENTS I take this opportunity to express my deepest gratitude to my supervisor Sudipta Mukherji for his unwavering support, guidance, and help during every stage of these projects. Without his guidance and generous help, it would not have been possible to complete these works and the thesis. I have learned many things from him which shaped my knowledge and thinking in physics. I wish to acknowledge my sincere gratitude to Amitabh Virmani as a collaborator and friend. I have been benefited hugely from him in many ways by several stimulating and fruitful discussions. I am also thankful to my other collaborators and group members at IOP, in particular, Sudipto Paul Chowdhury, Suman Ganguli, Samrat Bhowmick, Souvik Banerjee, Yogesh K. Srivastava, Shamik Banerjee, Bidisha Chakrabarty, Pratik Roy, Sumit Nandi for various stimulating discussions. In this regard, I want to mention my special thank to Sudipto Paul Chowdhury for helping me in Mathematica and in Latex. I am also thankful to DAE, Govt. Of. India and DST-Max Planck Partner Group Quantum Black Holes between IOP Bhubaneswar and AEI Golm for supporting me financially during my stay at IOP. I would like to thank my friends, teachers and other members of IOP for many help and making my stay here enjoyable and memorable one. Finally, I wish to thank my parents for their unconditional love, support, and encouragement during my Ph.D. days. They remain and shall be my source of inspiration at each and every moment of my life. List of Publications Arising From The Thesis 1. ”A note on AdS cosmology and gauge theory correlator” Souvik Banerjee, Samrat Bhowmick, Soumyabrata Chatterjee, Sudipta Mukherji JHEP 1506(2015),043 [arxiv:1501.06317] 2. ”Non-Vacuum AdS Cosmology and Comments on Gauge Theory Correlator” Soumyabrata Chatterjee, Sudipto Paul Chowdhury, Sudipta Mukherji, Yogesh K. Srivas- tava Phys. Rev. D 95, 046011 (2017)[arxiv:1608.08401] 3. ”Towards Timelike Singularity Via AdS Dual” Samrat Bhowmick, Soumyabrata Chat- terjee Int. J. Mod. Phys. A 32, 1750122 (2017)[arxiv:1610.05484] Talks Given In Conferences 1. “AdS cosmology and gauge theory correlator” National String Meeting(NSM 15), December, 2015, IISER Mohali, India. 2. “Non vacuum AdS cosmology and gauge theory correlator” Indian String Meeting(ISM 16), December, 2016, IISER Pune, India. Contents SYNOPSIS 13 1 Introduction 19 1.1 D-branes and supergravity solutions . 22 1.1.1 D-brane . 22 1.1.2 D-Branes as Solitonic Solutions of Supergravity . 23 5 1.1.3 AdS 5 × S from the Near Horizon Limit . 26 1.1.4 Matching of Symmetries: Towards a Duality . 27 1.1.5 The ’t Hooft Limit . 28 1.1.6 The AdS/CFT Dictionary and Recipe For Calculating Boundary Cor- relation Functions . 29 1.1.7 Calculation of the Two Point Function . 30 1.2 Some Aspects of Quantum Field Theory on Time Dependent Spacetime . 34 1.2.1 Field Theory on Milne Spacetime . 39 1.3 Plan of The Thesis . 42 2 Time Dependent Geometries and Their Duals 45 2.1 Kasner, AdS-Kasner, AdS-Kasner Soliton Geometries . 46 2.2 Supergravity Origin Of AdS-Kasner and AdS-Kasner Soliton Geometry . 50 2.2.1 AdS-Kasner From D3 Brane . 50 9 10 CONTENTS 2.2.2 AdS-Kasner Soliton From M5 Brane . 51 2.3 Other Time Dependent Duals in the Presence of Matter . 55 2.4 Flat Space Embedding of The Dual Geometries . 62 2.4.1 AdS-FRW embedding . 63 2.4.2 Embedding AdS-Kasner . 65 2.4.3 AdS-Kasner soliton . 68 2.5 Summary . 69 3 Gauge Theory Correlators On Time Dependent Spacetimes 71 3.1 AdS-Milne spacetime: An exactly solvable Model . 72 3.2 Problems in Calculating Correlators For The Others Geometries . 74 3.3 Geodesic Approximation . 77 3.4 Correlators in the Geodesic Approximation . 79 3.4.1 AdS - Kasner . 79 3.4.2 AdS-Kasner Soliton . 88 3.5 Non Singular Behaviour And Comparison With Other Results In The Literature 93 3.6 Appendix . 94 4 Space dependent AdS-Kasner 99 4.1 D3 brane with timelike Kasner world volume . 100 4.2 Investigating Timelike Singularity . 102 4.2.1 Extremal Space-like Co-dimension 4 Curve in bulk . 102 4.2.2 Extremal Space-like Co-dimension 2 Surface in bulk . 110 4.3 Discussion . 113 4.4 Appendix . 114 5 In Lieu of a Conclusion 117 CONTENTS 11 Bibliography 119 SYNOPSIS While the conventional perturbative approaches are well suited for the weakly coupled field theories, these methods fail when the theories become strongly coupled. One therefore needs to search for other avenues to explore the strongly coupled phases of field theories. Fortunately, the discovery of AdS/CFT correspondence of Maldacena in 1997 [1] has opened up one such direction. This correspondence relates a d spacetime dimensional gauge theory to a gravitational theory in d + 1 spacetime dimensions [2, 3]. One of the striking features of this approach is that it is inherently non-perturbative. This is in the sense that when one theory is weakly coupled, the other is in its strongly coupled phase. Consequently, while a direct proof of this conjecture still eludes us, it has passed through several non-trivial checks by now. Among the various forms of this correspondence, particularly well studied one, and mainly of interest to us in this thesis, is known as the weak form of the duality. In this form, the conjecture states that the type IIB supergravity on AdS 5 × S 5 is dual to the N = 4 large N, SU(N) gauge theory, at zero temperature, formulated on a four-dimensional flat spacetime. The gauge theory under consideration is strongly coupled in the sense that the ’t Hooft coupling is large. The gauge theory at finite temperature can also be studied within this framework by introducing a black hole in the gravity side in such a way that the asymptotic symmetries of AdS 5 remain unchanged. In this thesis, our primary aim is to explore some features of this gauge theory (and some cousins of it), formulated on time dependent backgrounds. Time dependent geome- 13 14 CONTENTS tries typically suffer from space-like singularities. Consequently, one expects to encounter strong gravitational fluctuations close to these singularities and, therefore, invalidating any semiclassical approach to study field theory on generic time dependent backgrounds. Even if one is probing away from these strongly fluctuating regions, because of the lack of the time-translational symmetry, defining a vacuum in a field theory on such a geometry becomes ambiguous. This appears to be even more serious when the spacetime possesses no static in and out regions [9]. It may therefore seem that there is very limited scope of learning anything about strongly coupled theories on time dependent backgrounds. Though, in general, it is true, some progress can indeed be made for geometries which possess AdS duals [14,28,29]. In this thesis we construct several four dimensional time dependent spacetimes with five dimensional AdS duals (besides those which are already known) and make a general attempt to study gauge theories on these geometries. In the next few paragraphs we summarize our works and the findings. From the evidences coming from the thermal background radiation, we know that our universe is isotropic on large scales. In order to uncover the reasons for such a high degree of isotropy, physicists, even as early as in 1970, asked the following question: Could it be that the particle creation in an initial anisotropic universe may back-react on the geometry, making it isotropic [30]? One such anisotropic background, belonging to the Bianchi I class, is the Kasner metric [31]1.