Boundary Conditions for Warped AdS$_3$ in Quadratic Ensemble

TL;DR

This paper introduces new boundary conditions in three-dimensional TMG that support a dual WCFT in quadratic ensemble, enabling derivation of warped Cardy formula and analysis of warped BTZ black hole entropy.

Contribution

It finds a novel set of boundary conditions in TMG that realize a WCFT in quadratic ensemble, facilitating the derivation of thermodynamic properties of warped black holes.

Findings

Derived a closed-form metric with two chiral functions.

Obtained the bulk thermodynamic entropy using warped Cardy formula.

Identified conditions for the vacuum state in the solution space.

Abstract

In the context of warped conformal field theories (WCFT), the derivation of the warped Cardy formula relies on the zero mode spectrum being bounded from below. Generically, this is not true for holographic WCFTs in "canonical" ensemble, whereas this condition is satisfied in the "quadratic" ensemble, making it more natural in holographic setups. In this work, we find a new set of boundary conditions in three-dimensional Topologically Massive Gravity (TMG) such that the putative dual theory is a WCFT in quadratic ensemble. Surprisingly, imposing the equations of motion, we obtain a closed form metric spanned by two arbitrary chiral functions, analogous to the Ba\~nados metrics in Einstein gravity. Surface charges for these boundary conditions are not a priori integrable and we discuss two choices of boundary conditions to make them so. We obtain the bulk thermodynamic entropy of warped BTZ black holes by making use of the warped Cardy formula, in its regime of validity. We also discuss the issue of identifying the vacuum in our solution space: demanding the enhancement of global symmetries selects only a family of solutions, out of which the unique vacuum must be carefully singled out.