Symplectic Geometry of Character Varieties and SU(2) Lattice Gauge Theory I

Associated to any finite graph \(\Lambda \) $\mathrm{\Lambda }$ is a closed surface \({\textbf{S}}={\textbf{S}}_\Lambda \) $\mathbf{S}={\mathbf{S}}_{\mathrm{\Lambda }}$ , the boundary of a regular neighbourhood of an embedding of \(\Lambda \) $\mathrm{\Lambda }$ in any three manifold. The surface retracts to the graph, mapping loops on the surface to loops on the graph. The (SU(2)) character variety \({{\mathcal {M}}}\) $\mathcal{M}$ of \({\textbf{S}}\) $\mathbf{S}$ has a symplectic structure and associated Liouville measure; on the other hand, the character variety \({\textbf{M}}\) $\mathbf{M}$ of \(\Lambda \) $\mathrm{\Lambda }$ carries a natural measure inherited from the Haar measure. Loops on \({\textbf{S}}\) $\mathbf{S}$ define functions on the character varieties, the Wilson loops. By the works of W. Goldman, L. Jeffrey and J. Weitsman, the formalism of Duistermaat-Heckman applies to the relevant integrals over \({{\mathcal {M}}}\) $\mathcal{M}$ . We develop a calculus for calculating correlations of Wilson loops on \({{\mathcal {M}}}\) $\mathcal{M}$ w.r.to the normalised Liouville measure, and present evidence that they approximate—for large graphs—the corresponding integrals over \({\textbf{M}}\) $\mathbf{M}$ . Lattice field theory involves integrals over \({\textbf{M}}\) $\mathbf{M}$ ; we present “symplectic” analogues of expressions for partition functions, Wilson loop expectations, etc., in two and three space-time dimensions.