Maxwell Chern-Simons gravity in 3D: Thermodynamics of cosmological solutions and black holes with torsion

TL;DR

The paper develops a coherent thermodynamic framework for three-dimensional Maxwell Chern–Simons gravity and its torsionful extension. By introducing generalized asymptotic conditions with chemical potentials, it demonstrates that solutions carry mass, angular momentum, and a global spin-2 charge, and derives a consistent entropy formula extending the Bekenstein–Hawking result with explicit spin-2 contributions. The torsionful model, built from a deformed Maxwell algebra, yields BTZ-like black holes with non-Riemannian geometry and an extended asymptotic symmetry algebra isomorphic to \mathfrak{bms}_3\oplus\mathfrak{vir}, with thermodynamics computed via holonomy regularity and first laws verified. In both sectors, the entropy can be expressed as a reparametrization-invariant integral of the horizon’s induced spin-2 fields, highlighting the role of the Maxwell field and torsion in gravitational thermodynamics. The results connect flat and torsional three-dimensional gravity to a broader Maxwellian framework and suggest several avenues for future work, including non-relativistic limits and supersymmetric extensions.

Abstract

We construct generalized sets of asymptotic conditions for both three-dimensional Maxwell Chern-Simons gravity and a novel extension that incorporates torsion through a deformation of the Maxwell algebra. These boundary conditions include the most general temporal components of the gauge fields that consistently preserve the corresponding asymptotic Maxwell algebras with identical classical central charges, while allowing for the inclusion of chemical potentials conjugate to the conserved charges. We show that both sets of asymptotic configurations admit nontrivial solutions carrying not only mass and angular momentum but also an additional global spin-2 charge. In the torsionless case, the theory admits locally flat cosmological spacetimes, whereas in the presence of torsion, it generalizes to BTZ-like black hole geometries. For each case, the thermodynamic properties are consistently derived in terms of the gauge fields and the topology of the Euclidean manifold, shown to correspond to a solid torus. Furthermore, we obtain a general expression for the entropy, depending on both the horizon area and its spin-2 analogues, which can be written as a reparametrization-invariant integral of the induced spin-2 fields on the spacelike section of the horizon.