Topologically massive gravity and its conformal limit

Abstract

Three dimensional gravity has been known for some time to be a playground for testing ideas and problems of higher dimensional gravitational theories. Nevertheless its status as a toy model for quantum gravity is still uncertain.<br />Already in 1986 Brown and Henneaux discovered that three dimensional quantum gravity with negative cosmological constant is dual to a two dimensional conformal field theory (CFT) in the sense that the Hilbert space must fall into unitary representation of two copies of the Virasoro algebra.<br />They obtained, in quantizing this theory, an asymptotic Virasoro algebra with central charges c_L=c_R=(3 l)/(2 G_N), where G_N is Newton's constant and $\ell$ parameterizes the cosmological constant.<br />Almost ten years later black hole solutions for this three dimensional theory were discovered by Banados, Teitelboim and Zanelli. In the same period of time further milestones of relevance for this work have been established: the AdS/CFT correspondence by Maldacena in 1997 and the proposal by Witten in 2007 to define three dimensional quantum gravity in terms of its dual CFT.<br />Over the last few years many attempts have been made to construct gravitational theories in three dimensions that could serve as toy models for quantum gravity. Since a pure Einstein-Hilbert action with a negative cosmological constant lacks additional degrees of freedom one can remedy this by adding a gravitational Chern-Simons term. This results in a theory that exhibts black holes and gravitons and is called topologically massive gravity (TMG).<br />The first part of this thesis deals with finding exact solutions of TMG.<br />This is an interesting problem already at the classical level since non-trivial solutions to the equations of motion are hard to find and only few are known. An efficient way to find solutions is to dimensionally reduce the theory by using two commuting Killing vectors. This results in a (0+1)-dimensional model in which it is then possible to classify all stationary axi-symmetric solutions of the three dimensional counterpart. Besides this classification} and the construction of suitable numerical algorithms the most intriguing and new results are solitonic solutions that show asymptotic warped AdS behaviour. More precisely, they show damped oscillations around warped AdS.<br />Then emphasis is put on the conformal limit of TMG leading to a theory called conformal Chern-Simons gravity. Motivated by partial masslessness, which provides an additional gauge symmetry, a specific set of boundary conditions is chosen. This specific set comprises boundary conditions on the conformal class of the metric and the Weyl factor. A complete holographic analysis, including calculations of the boundary stress tensor and the canonical charges, gives rise to interesting features of the dual CFT.<br />Depending on the boundary conditions on the Weyl factor the CFT has different properties. For fixed Weyl factor the central charges are c_R=-c_L=12k. For varying Weyl factor the dual CFT contains a scalar field with background charge resulting in a shifted value for the left central charge -c_L=12k+1+6Q 2.