Thermodynamics of Chern-Simons AdS$_5$ black holes coupled to $\mathrm{SU}(2)$ solitons

TL;DR

This paper studies five-dimensional Chern--Simons AdS black holes coupled to SU(2) solitons, analyzing their thermodynamics, conserved charges, and entropy contributions of torsion parameters using a minisuperspace approximation.

Contribution

It introduces a variational framework for these black holes that captures boundary terms, conserved quantities, and reveals the role of torsion parameters in entropy.

Findings

Recovered known energy and U(1) charge using boundary terms.

Identified a momentum conjugate to trace-torsion mode.

Found torsion parameters contribute nontrivially to black hole entropy.

Abstract

We investigate properties of the five-dimensional Chern--Simons AdS black hole coupled to $\mathrm{SU}(2)$ solitons by means of a minisuperspace approximation adapted to static, spherically symmetric configurations. The reduced action reproduces the known branch of solutions and provides a variational framework in which the boundary terms determine the conserved quantities and their conjugate variables. In particular, we recover the energy and the $\mathrm{U}(1)$ charge previously obtained by Hamiltonian methods, while the enlarged parameter space also reveals a momentum conjugate to the trace-torsion mode. The Euclidean action yields an entropy satisfying the first law of black hole thermodynamics. In contrast to many other torsional black hole models, the axial torsion parameter, describing the secondary black hole hair, together with the trace-torsion mode, contributes nontrivially to the entropy. The expression for the entropy obtained in this way is further confirmed by the other two methods found in the literature.