Three Dimensional Gravity From SU(2) Yang-Mills Theory in Two Dimensions

TL;DR

This paper demonstrates that two-dimensional SU(2) Yang-Mills theory encodes the geometry of Riemann surfaces embedded in three-dimensional curved spaces, linking gauge theory to three-dimensional gravity and integrable models.

Contribution

It establishes a novel correspondence between 2D SU(2) Yang-Mills theory and the embedding of Riemann surfaces in 3D manifolds, revealing new insights into gravity and integrable systems.

Findings

Yang-Mills field strength encodes ambient space curvature

Solutions to Gauss-Codazzi equations relate to integrable models

3D Chern-Simons theory describes surface dynamics in 4D spacetime

Abstract

We argue that two dimensional classical SU(2) Yang-Mills theory describes the embedding of Riemann surfaces in three dimensional curved manifolds. Specifically, the Yang-Mills field strength tensor computes the Riemannian curvature tensor of the ambient space in a thin neighborhood of the surface. In this sense the two dimensional gauge theory then serves as a source of three dimensional gravity. In particular, if the three dimensional manifold is flat it corresponds to the vacuum of the Yang-Mills theory. This implies that all solutions to the original Gauss-Codazzi surface equations determine two dimensional integrable models with a SU(2) Lax pair. Furthermore, the three dimensional SU(2) Chern-Simons theory describes the Hamiltonian dynamics of two dimensional Riemann surfaces in a four dimensional flat space-time.