Torsion as a Gauge Field in a Lorentz-Chern-Simons Theory

TL;DR

This paper presents a gravity model in 2+1 dimensions where torsion is interpreted as a gauge field, leading to a unification of electromagnetism and gravity within a Riemann-Cartan geometric framework.

Contribution

It introduces a novel interpretation of torsion as a gauge field for the Weyl group, connecting gravity and electromagnetism through a Chern-Simons formulation in 2+1 dimensions.

Findings

Torsion's trace acts as a gauge field for Weyl symmetry.

The model reduces to Chiral Gravity coupled with a Chern-Simons-Proca theory.

Electrodynamics and gravity are unified as geometric features of a Riemann-Cartan manifold.

Abstract

We explore a model of gravity that arises from the consideration of the Chern-Simons form in 2+1 dimensions for a spin connection with a contorsion described by a scalar and a vector field. The effective Lagrangian presents a local Weyl symmetry allowing us to gauge the scalar field to a constant value. From a gauge field theory perspective, it is shown that the vector part of the torsion (related to its trace) is a gauge field for the Weyl group, which allows the interpretation of the torsion as an electromagnetic field. In the gauge of constant scalar field we obtain Chiral Gravity coupled to a Chern-Simons-Proca theory for the vector field, that at the level of equations of motion is equivalent to Topologically Massive Electrodynamics minimally coupled to Chiral Gravity. Electrodynamics and gravity appear here unified as geometrical features of a Riemann-Cartan manifold.