## The first law of thermodynamics and 2d CFT descriptions for near-extremal and near-EVH black holes

##### Abstract

In this thesis we investigate the quantum aspects of black holes near extremality. In particular
we seek evidence that a near-extremal black hole has a microscopic description in terms of a
two dimensional conformal field theory (CFT).
We first demonstrate how the low temperature expansion of the first law of thermodynamics
leads to an expression for the entropy of extremal black holes which can be recast as the Cardy
formula for the entropy of a chiral two dimensional CFT, in agreement with the Extremal Black
Hole/CFT correspondence. We apply Sen’s entropy function formalism to fortify this result by
reproducing it in a gravitational setup.
We extend our first law analysis to a class of near-Extremal Vanishing Horizon (near-EVH)
black holes. These black holes have low entropy and temperature, and their geometries contain
locally asymptotically AdS3 throats in the near horizon region. The low temperature expansion
of the first law is compatible with the first law for a three dimensional BTZ black hole. As the
BTZ black hole has an AdS/CFT description in terms of a non-chiral two dimensional CFT,
our result can be viewed as thermodynamic evidence for the EVH/CFT correspondence, which
states that gravity on the near horizon EVH geometry is described by a 2d CFT. A near-EVH
black hole, or low energy excitation around an EVH black hole, is described by excitations of
the dual 2d CFT.
As case studies of our first law analysis and the EVH/CFT correspondence, we focus on
two asymptotically AdS5× S5 classes of near-EVH black holes. The two cases have interesting
individual properties and, by the AdS/CFT correspondence, dual descriptions as states in
N = 4 super Yang-Mills theory . We can compare these (UV) pictures to the two dimensional
descriptions that emerge from the near horizon, or low energy, dynamics. All EVH near horizon
geometries have local AdS3 factors which become BTZ black holes when the configurations are
excited from EVH to near-EVH.
In the study of static black holes with three R-charges, we examine the non-BPS and near-
BPS regimes separately. While the non-BPS near horizon limit is locally regular, in the near-
BPS case the near horizon procedure requires focussing geometrically on a strip of the horizon,
and the degrees of freedom of the dual CFT2 can be associated with stretched strings between
giant gravitons in the transverse five-sphere.
The near-EVH limit of non-BPS stationary charged black holes is obtained by taking the
vanishing limit of one or both of the angular momenta. When one of the momenta is small, the
AdS3 angle is a combination of azimuthal angles in the AdS5 and S5 regions of the geometry.
Taking the vanishing limit of both of the angular momenta leads to a near horizon limit which
contains a BTZ black hole that is non-trivially fibred by a three-sphere.
For each of the case studies we use the AdS3/CFT2 dictionary to specify dual IR CFT2
descriptions of the black holes. We outline a map between the UV and IR near-EVH excitations
and demonstrate how the first law reduces in the near-EVH limit to the first law of a
BTZ black hole. As a consistency check we compare our results with those of the Kerr/CFT
correspondence.