Aspects of conformal field theories at finite temperature

Abstract

In this thesis we have studied broadly two aspects of thermal field theory. We began by examining how the macroscopic system (described by relativistic hydrodynamics) behalves in presence of microscopic anomalies. We are able to relate macroscopic transport coefficients to the anomalous conservation equations of the microscopic theory. It is to be noted that, using the perturbative methods that we develop, we are able to relate both the mixed and pure gravitational anomalies to their respective transport coe fficients. Our results agree with other methods used to study this relationship. Using our perturbative approach, we are also able to understand the breakdown of the replacement rule for gravitino systems. Global anomalies instead of perturbative anomalies can also be used to x the macroscopic transport coefficients. By computing the global anomalies associated with particular systems, we were able to write down thermal effective actions which reproduce the anomalies. We show that such effective actions can be used to compute the transport coefficients and obtain a match with our perturbative results. We also provide a topological understanding of the replacement rule. As a further check of our formalism, we compute perturbatively using the formalism developed in [11], the anomalous transport coefficient (corresponding to pure gravitational anomaly) for self dual tensors in d = 6 and obtain a match with the global anomaly result. In the second part of the thesis we look at constraints that can be placed on spectral densities in a conformal field theory at fi nite temperature. Sum rules provide important constraints on spectral densities of any quantum field theory. We relate the weighted integral of spectral densities over frequency to the energy density of the theory. We show that the proportionality constant can be written down in terms of Hofman-Maldacena variables t2 and t4, which determine the three point function of stress tensors of a parity preserving CFT. For CFTs dual to two derivative Einstein gravity, we nd agreement of our sum rule derived from general conformal invariance with holographic methods. We also obtain correction to the holographic shear sum rule for theories with quadratic curvature corrections to the Einstein gravity. We extend the conformal collider physics formalism developed by Maldacena et al to study three point functions involving a stress tensor T, a U(1) current j, in 2 + 1 dimensional parity violating conformal field theories. We show that large N Chern Simons theories coupled to fundamental fermions/ bosons saturate our derived bounds. This is consistent with the observations that the scaling dimensions of spin operators in these theories saturate the unitarity bound ( s s + 1) and hence perhaps the conformal collider bounds as well.