Generating Jackiw-Teitelboim Euclidean gravity from static three-dimensional Maxwell-Chern-Simons electromagnetism

We consider pure three-dimensional Maxwell-Chern-Simons electrodynamics in the static limit. We show that this theory can be mapped onto a two-dimensional gravitational model in the first-order formalism of Riemannian manifolds with Euclidean signature, coupled to a real scalar field naturally interpreted as a dilaton. In this framework, the Newtonian and cosmological constants in two dimensions are fully determined by the electric charge. The solution to this gravitational model is found to be trivial: a constant dilaton field on a flat manifold. However, we introduce two distinct shifts of the spin-connection that transform the model into Jackiw-Teitelboim gravity. Specifically, we identify two additional solutions: a hyperbolic manifold with also a constant dilaton configuration; and a spherical manifold where, again, the dilaton assumes a constant, nonzero field configuration. In both non-flat cases, by employing the Gauss-Bonnet theorem in the specific cases of compact manifolds, we establish that the manifold's radius is fixed by the cosmological constant (and, therefore, by the electric charge).