Symplectic Geometry of character varieties and $SU(2)$ Lattice Gauge Theory I

TL;DR

This paper explores the symplectic geometry of $SU(2)$ character varieties related to lattice gauge theories, providing new formulas for partition functions and correlations, and connecting geometric and quantum field theoretic perspectives.

Contribution

It introduces a symplectic geometric framework for analyzing $SU(2)$ lattice gauge theories and derives explicit formulas for partition functions and observables.

Findings

Partition functions expressed as positive series sums

Connections established between geometry and lattice gauge theory

Explicit formulas for correlations and 't Hooft loops

Abstract

Given a finite connected graph $\Lambda$, the space of $SU(2)$ lattice gauge-fields on $\Lambda$, modulo gauge transformations, is a Lagrangian submanifold -- with mild singularities -- of the $SU(2)$ character variety (= phase-space of Chern-Simons theory) of an associated surface. We present evidence that, in the limit of large $\Lambda$, integration over the character variety with respect to the Liouville measure approximates lattice-theoretic integrals. By the works of W. Goldman, L. Jeffrey and J. Weitsman, the formalism of Duistermaat-Heckman applies to the relevant integrals over the character variety. A continuous version of the Verlinde algebra facilitates computations. In two dimensions we recover standard expressions. For the theory on a 3-dimensional periodic lattice $\Lambda$ with Migdal action we get a very pleasant expression for the symplectic partition function, and with the Wilson action a more elaborate one. Each is a sum of a series with positive terms. One can also write down expressions for plaquette-plaquette correlations and 't Hooft loops.