A Study of Symmetries and Phases in Gravity with Application to Holography and Cosmology A thesis submitted to Tata Institute of Fundamental Research, Mumbai, India for the degree of Doctor of Philosophy in Physics By Nilay Kundu Department of Theoretical Physics Tata Institute of Fundamental Research Mumbai - 400 005, India September, 2014 (Final Version Submitted in May, 2015) Declaration This thesis is a presentation of my original research work. Wherever contribu- tions of others are involved, every effort is made to indicate this clearly, with due reference to the literature, and acknowledgement of collaborative research and discussions. The work was done under the guidance of Professor Sandip P Trivedi, at the Tata Institute of Fundamental Research, Mumbai. (Nilay Kundu) In my capacity as the supervisor of the candidate’s thesis, I certify that the above statements are true to the best of my knowledge. (Sandip P Trivedi) Acknowledgments Firstly, I am deeply grateful to my thesis supervisor Sandip P Trivedi for his constant support and encouragement. His insights and intuition for Physics has been a great source of learning for me. The lengthy discussion sessions with him, as I believe, were the most enjoyable part of my days as a research scholar at TIFR. His enormous enthusiasm and excitement for research as well as simple but insightful ways of explaining things to students as a teacher has been immensely influential and motivating. I would also like to thank Shiraz Minwalla, Gautam Mandal for many stimulating discussions and also for the wonderful courses they taught. I would also like to thank all the other faculty members of string group for many stimulating discussions and comments. I am grateful to the other faculty members of our department, specifically to Sunil Mukhi, Deepak Dhar, Saumen Datta, Kedar Damle for teaching great courses during my course work at TIFR. I would also like to thank my childhood teachers Mr. Abdul Halim and Pranati Di for all their support and inspirations. I am grateful to Prithvi Narayan, Nilanjan Sircar, Sachin Jain, Rickmoy Samanta and Ashish Shukla for wonderful collaboration and working together. Specifically, I would like to acknowledge the support and advice received from Prithvi Narayan which were very influential to have a smooth start of my research at TIFR. I was also fortunate to have seniors like Jyotirmoy, Satyabrata and Sayantan for their kind help in many ways. Over the years I have stayed at TIFR through all the ups and downs, the theory students room with all its inmates has been the most comfortable place to stay. I would like to thank all the other students in DTP, who were present during the course of my stay, for maintaining such an excellent and lively environment. It was great to come across Debjyoti, Geet, Kusum, Mangesh, Nikhil, Padmanath, Rahul, Ritam, Sambuddha, Tarun, Umesh, Yogesh and also the other friends of mine at TIFR, especially Amaresh, Chandradoy, Kinjalk, Naveen, Pankaj, Sanmay, Sumanta and Vivek. It was great fortune to have them in all the fun during the weekend football, food outside or be it the monsoon treks in Sahyadri. I am also grateful to my friends Manik, Devagnik and Pinaki from my university days at Jadavpur for their support. It would be otherwise incomplete if I don’t admit that I owe a lot to Sudeshna for this and that and mostly for being there all along as a friend. Finally and above all, this thesis owes a lot for its existence to the unconditional support I got from my parents and sister. Undoubtedly, it is my parents, all their sacrifices and compromises in their own lives, along with their encouragement and belief in me, which have driven me to pursue my research. Collaborators This thesis is based on work done in collaboration with several people. The work presented in chapter 2 was done in collaboration with Norihiro Iizuka, • Shamit Kachru, Prithvi Narayan, Nilanjan Sircar and Sandip P. Trivedi and is based on the publication that appeared in print as JHEP 1207 (2012) 193. The work presented in chapter 3 was done in collaboration with Prithvi Narayan, • Nilanjan Sircar and Sandip P. Trivedi and is based on the publication that appeared in print as JHEP 1303 (2013) 155 The work presented in chapter 4 was done in collaboration with Archisman Ghosh, • Suvrat Raju, and Sandip P. Trivedi and is based on the publication that appeared in print as JHEP 1407 (2014) 011. To My Baba and Ma Synopsis Introduction One of the most challenging and conceptually deep problems in theoretical physics, in recent years, has been to understand the quantum nature of gravity. String theory, besides being a consistent and mathematically elegant framework in itself, is considered to be the leading candidate to demystify quantum gravity. It contains a plethora of low energy ground states, usually referred to as the landscape. Each one of these vacua has widely different symmetry properties from others and the low energy excitations around each one of them constitute different phases of the theory. These different phases emerging out of String theory are able to explain the familiar symmetries that play a role in nature. It contains, within its framework, the ingredients to realize, for example, the symmetries of the Standard Model of particle physics. In addition, String theory embodies new and strange symmetries beyond the Standard model, e.g. supersymmetry, which is ubiquitous in it’s construction but is yet to be realized in experiments. Therefore it is of immense importance to study and understand symmetries and phases in general within the framework of String theory. In this thesis we explore various aspects of symmetries and study different phases in a gravity theory to understand some interesting features prevalent in nature around us. In particular, guided by the lessons from such studies in gravity we find their applications to two considerably different areas of physics: strongly coupled field theories and early Universe cosmology. To establish this connection, the main computational technique that we use, follows from the AdS/CF T 1 correspondence in String theory, also known as holography. The AdS/CF T correspondence, also frequently referred to as gauge-gravity duality, is one of the most significant results that String theory has produced. It relates two considerably different theories, a gauge theory without gravity to a quantum theory of gravity living in one higher dimension, see [1] for a review and the references therein. Most importantly, being a strong-weak duality it provides us a powerful computational tool to explore strongly coupled field theories, which are otherwise beyond the scope of perturbative calculation, via tractable Einstein gravity. As a consequence, research in String theory has now found its widespread application in other branches of physics as well, for example, condensed matter 1The abbreviation AdS/CFT stands for anti de-Sitter/conformal field theory. ix Synopsis physics, QCD etc.; see the reviews [2, 3, 4, 5] for connections to condensed matter physics and [6] for QCD related applications. On the other hand, both theoretically and observationally, the physics of our Universe shortly after its birth is by itself a fascinating subject of modern day research. The approximate homogeneity and isotropy at large enough length scales leads us to believe that our Universe at that early stage went through a tiny phase of rapid expansion. In cosmology, theoretical calculations on various toy models come up with predictions that are tested against observations. In view of the abundance of models and lack of any compelling one it is more natural to try to understand model independent features of the early Universe based on symmetry considerations alone. Interestingly, as we will see in this thesis, in some contexts for the study of early Universe the holographic principle happens to be an useful computational tool. In the first part of the thesis, based on [7], we address the question: what kind of phases found in nature can be realized in gravity description. With the help of a symmetry classification for the generalized translational symmetries, named the Bianchi classification, we find possible near horizon gravity solutions, which are homogeneous but not necessarily isotropic, falling into each class. These gravity solutions extend the existing set of a very few solutions known so far on the gravity side and correspond to similar homogeneous but possibly anisotropic phases abundantly observed in condensed matter systems. In the next part of the thesis, based on [8], we run our investigation the other way round and try to understand if phases in gravity are of interest in condensed matter physics. We consider a specific solution in a system of gravity coupled to scalar and gauge fields with a negative cosmological constant. It corresponds to a compressible phase on the field theory side with an unbroken global symmetry at finite chemical potential, e.g. Fermi liquids. We further investigate the existence of a Fermi surface in those phases by looking at its response to turning on a small magnetic field. For that we study an interesting property of the system called entanglement entropy and compute it using holography in the gravity side. In the third and concluding part of this thesis, in [9], we study conformal invariance during inflation, a proposed theoretical mechanism to describe the phase of rapid expansion of our Universe at a very early epoch. The approximate de-Sitter (dS4) spacetime during inflation has symmetries which are same as that of a 3 dimensional Euclidean conformal field theory (CFT) and that imposes non-trivial constraints on the correlation functions of the cosmological perturbations produced at that time. These correlation functions are measured directly in the sky by looking at the anisotropies in the cosmic microwave background (CMB) radiation.