Partition Functions in AdS/CFT

Abstract

Quantum field theory has been a useful framework for analyzing and understanding the fundamental forces of nature, including electromagnetism, weak and strong interactions. However, when one tries to apply the usual techniques of quantizing a field theory to gravity, it does not yield a sensible outcome. The Anti-de Sitter (AdS)/Conformal Field Theory (CFT) correspondence has been of great importance among the alternative approaches to defining a consistent quantum gravity theory. AdS/CFT states that gravitational theories with a negative cosmological constant in d + 1 dimensions are equivalent to a special class of quantum field theories called conformal field theories in d dimensions with no gravitational dynamics. The exact equivalence between the two theories requires a mapping that tells us how physical quantities in one theory are related to physical quantities in the other theory. One can check this proposed equivalence by computing some of these physical quantities like the correlation functions and partition functions in gravity and the conformal field theory. In this project, we focus mainly on later of these objects for three-dimensional gravity and the corresponding dual two-dimensional CFT.

After an introduction in chapter 1 regarding the notation and basics of the subject, in chapter 2, we begin by investigating the known computations and results in this direction for AdS3 gravity. We then move to the analogous computations for CFT2 in chapter 3. In the process, we will encounter the problems one faces while assigning a physical interpretation to these results. Subsequently, we then study its proposed resolutions and their physical implications. To better understand and address the problem, in chapters 4 and 5, we consider the study of gravity partition function in the context of Rational CFTs. In these cases, some of the calculations are simplified. Specifically, we study the SU(N) level k WZW models and Virasoro minimal models in great detail. We investigate its relation to the recently proposed idea about an ensemble of CFTs dual for AdS quantum gravity in three dimensions.