Three-dimensional gravity from SU(2) Yang-Mills theory in two dimensions

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.