# Log corrections to entropy of three dimensional black holes with soft hair

## TL;DR

This paper computes log corrections to the entropy of three-dimensional black holes with soft hair, using two approaches that relate different boundary conditions and symmetries, revealing consistent results across methods.

## Contribution

It introduces two methods to evaluate log corrections for black holes with soft hair, connecting boundary conditions and dual theories with anisotropic scaling.

## Key findings

- Log corrections for BTZ and soft hairy black holes agree for all non-negative z.
- Log corrections depend only on zero modes for soft hairy boundary conditions.
- Results unify thermodynamic corrections across different boundary conditions.

## Abstract

We calculate log corrections to the entropy of three-dimensional black holes with "soft hairy" boundary conditions. Their thermodynamics possesses some special features that preclude a naive direct evaluation of these corrections, so we follow two different approaches. The first one exploits that the BTZ black hole belongs to the spectrum of Brown-Henneaux as well as soft hairy boundary conditions, so that the respective log corrections are related through a suitable change of the thermodynamic ensemble. In the second approach the analogue of modular invariance is considered for dual theories with anisotropic scaling of Lifshitz type with dynamical exponent z at the boundary. On the gravity side such scalings arise for KdV-type boundary conditions, which provide a specific 1-parameter family of multi-trace deformations of the usual AdS3/CFT2 setup, with Brown-Henneaux corresponding to z=1 and soft hairy boundary conditions to the limiting case z=0. Both approaches agree in the case of BTZ black holes for any non-negative z. Finally, for soft hairy boundary conditions we show that not only the leading term, but also the log corrections to the entropy of black flowers endowed with affine u(1) soft hair charges exclusively depend on the zero modes and hence coincide with the ones for BTZ black holes.