Scalar solitons and the microscopic entropy of hairy black holes in three dimensions

TL;DR

This work analyzes 3D gravity with a self-interacting scalar, showing exact soliton and hairy black hole solutions under relaxed AdS boundary conditions. By identifying the scalar soliton as the ground state of the hairy sector, the authors reproduce the hairy black hole entropy microscopically via Cardy formula, using shifted Virasoro ground-state eigenvalues rather than central charges alone. The key insight is that two disconnected sectors (vacuum and hairy) require their own ground states, enabling a consistent holographic counting that matches the semiclassical entropy. The results hinge on the Virasoro structure with c = 3l/2G and a soliton-ground-state framework, with extensions to grand canonical and rotating cases discussed.

Abstract

General Relativity coupled to a self-interacting scalar field in three dimensions is shown to admit exact analytic soliton solutions, such that the metric and the scalar field are regular everywhere. Since the scalar field acquires slow fall-off at infinity, the soliton describes an asymptotically AdS spacetime in a relaxed sense as compared with the one of Brown and Henneaux. Nevertheless, the asymptotic symmetry group remains to be the conformal group, and the algebra of the canonical generators possesses the standard central extension. For this class of asymptotic behavior, the theory also admits hairy black holes which raises some puzzles concerning an holographic derivation of their entropy à la Strominger. Since the soliton is devoid of integration constants, it has a fixed (negative) mass, and it can be naturally regarded as the ground state of the "hairy sector", for which the scalar field is switched on. This assumption allows to exactly reproduce the semiclassical hairy black hole entropy from the asymptotic growth of the number of states by means of Cardy formula. Particularly useful is expressing the asymptotic growth of the number of states only in terms of the spectrum of the Virasoro operators without making any explicit reference to the central charges.

Figures (1)

• Figure 1: The asymptotic growth of the number of states can be written exclusively in terms of the spectrum of the shifted Virasoro operators $\tilde{L}_{0}^{\pm }$. Then, it is given by $\rho (\tilde{\Delta}^{+},\tilde{\Delta}^{-})=\rho (\tilde{\Delta}^{+})\rho (\tilde{\Delta}^{-})$, with $\rho (\tilde{\Delta}^{\pm })=\rho (\tilde{\Delta}_{0}^{\pm })\exp \left( 4\pi \sqrt{-\tilde{\Delta}_{0}^{\pm }\tilde{\Delta}^{\pm }}\right)$, where $\tilde{\Delta}_{0}^{\pm }$ and $\rho (\tilde{\Delta}_{0}^{\pm })$ correspond to the lowest eigenvalues of $\tilde{L}_{0}^{\pm }$ and their degeneracies, respectively.