Three-dimensional $\operatorname{SL}(2,\mathbb R)$ Yang-Mills theory is three-dimensional gravity with background sources

TL;DR

This paper establishes an equivalence between three-dimensional $ ext{SL}(2, ext{R})$ Yang-Mills theory and a form of three-dimensional gravity coupled to a background source, extending topological Chern-Simons gravity to include dynamical degrees of freedom.

Contribution

It demonstrates that 3D $ ext{SL}(2, ext{R})$ Yang-Mills theory corresponds to a first-order gravity formulation with background sources, revealing new dynamical features beyond topological theories.

Findings

Yang-Mills theory's local degrees of freedom relate to degenerate gravitational waves.

Background stress-energy sources break diffeomorphism invariance.

Adding a cosmological constant yields new gauge theories with background sources.

Abstract

Chern-Simons theory with certain gauge groups is known to be equivalent to a first-order formulation of three-dimensional Einstein gravity with a cosmological constant, where both are purely topological. Here, we extend this correspondence to theories with dynamical degrees of freedom. We show that three-dimensional Yang-Mills theory with gauge group $\operatorname{SL}(2,\mathbb R)$ is equivalent to the first-order formulation of three-dimensional Einstein gravity with no cosmological constant coupled to a background stress-energy tensor density (which breaks the diffeomorphism symmetry). The local degree of freedom of three-dimensional Yang-Mills theory corresponds to degenerate "gravitational waves" in which the metric is degenerate and the spin connection is no longer completely determined by the metric. Turning on a cosmological constant produces the third-way (for $\Lambda<0$) or the imaginary third-way (for $\Lambda>0$) gauge theories with a background stress-energy tensor density.