The Semiclassical Limit for $SU(2)$ and $SO(3)$ Gauge Theory on the Torus

Abstract

We prove that for $SU(2)$ and $SO(3)$ quantum gauge theory on a torus, holonomy expectation values with respect to the Yang-Mills measure $dμ_T(ø) =N_T^{-1}e^{-S_{YM}(ø)/T}[{\cal D}ø]$ converge, as $T\downarrow 0$, to integrals with respect to a symplectic volume measure $μ_0$ on the moduli space of flat connections on the bundle. These moduli spaces and the symplectic structures are described explicitly.