Three-dimensional $SL(2,R)$ Yang-Mills theory is equivalent to three-dimensional gravity with background sources

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. As an example, we show that three-dimensional Yang-Mills theory with gauge group SLð2;RÞ 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 Λ < 0) or the imaginary third-way (for Λ > 0) gauge theories with a background stress-energy tensor density.