Yang-Mills on Surfaces with Boundary: Quantum Theory and Symplectic Limit

Abstract

The quantum field measure for gauge fields over a compact surface with boundary, with holonomy around the boundary components specified, is constructed. Loop expectation values for general loop configurations are computed. For a compact oriented surface with one boundary component, let \(\) be the moduli space of flat connections with boundary holonomy lying in a conjugacy class \(\) in the gauge group G. We prove that a certain natural closed 2-form on \(\), introduced in an earlier work by C. King and the author, is a symplectic structure on the generic stratum of \(\) for generic \(\). We then prove that the quantum Yang-Mills measure, with the boundary holonomy constrained to lie in \(\), converges in a natural sense to the corresponding symplectic volume measure in the classical limit. We conclude with a detailed treatment of the case \(\), and determine the symplectic volume of this moduli space.