Three-dimensional $\mathrm{SL}(2,\mathbb{R})$ Yang-Mills theory is three-dimensional gravity 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 $\mathrm{SL}(2,\mathbb{R})$ 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.

arXiv comment7 pages, added clarifying notes, published version

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