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

• Main Authors: Borsten, Leron, Kanakaris, Dimitri, Kim, Hyungrok
• Format: Journal Article
• Language: English
• Published: 26.08.2024
• Subjects: Physics - General Relativity and Quantum Cosmology; Physics - High Energy Physics - Theory
• DOI: 10.48550/arxiv.2408.14228

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.