Three Dimensional Gravity From SU(2) Yang-Mills Theory in Two Dimensions
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.
Cite
@article{arxiv.hep-th/0305168,
title = {Three Dimensional Gravity From SU(2) Yang-Mills Theory in Two Dimensions},
author = {Antti J. Niemi},
journal= {arXiv preprint arXiv:hep-th/0305168},
year = {2009}
}