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

Antti J. Niemi
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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.

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Antti J. Niemi. 2003-05-20. Three Dimensional Gravity From SU(2) Yang-Mills Theory in Two Dimensions. https://doi.org/10.1103/physrevd.70.045017

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