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

TL;DR

This paper constructs exact scalar soliton solutions in three-dimensional gravity with a scalar field, explores their role as ground states for hairy black holes, and successfully reproduces black hole entropy using the Cardy formula.

Contribution

It introduces regular scalar solitons as ground states in 3D gravity with scalar fields, enabling an exact microscopic derivation of hairy black hole entropy.

Findings

Exact scalar soliton solutions are regular everywhere.

The soliton acts as the ground state with fixed negative mass.

Black hole entropy matches the Cardy formula without explicit central charge reference.

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 \`a 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.