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:   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.

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