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

TL;DR

This paper proves that in quantum gauge theories on a torus, the expectation values of holonomies converge to integrals over the moduli space of flat connections as temperature approaches zero, linking quantum and classical descriptions.

Contribution

It establishes the semiclassical limit for $SU(2)$ and $SO(3)$ gauge theories on a torus, explicitly describing the moduli space and symplectic structure involved.

Findings

Holonomy expectation values converge to classical integrals as temperature decreases.

Explicit description of moduli space of flat connections.

Connection between quantum gauge theory and classical symplectic geometry.

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 $d\mu_T(\o) =N_T^{-1}e^{-S_{YM}(\o)/T}[{\cal D}\o]$ converge, as $T\downarrow 0$, to integrals with respect to a symplectic volume measure $\mu_0$ on the moduli space of flat connections on the bundle. These moduli spaces and the symplectic structures are described explicitly.