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

Abstract

Associated to any finite graph \(\Lambda \) is a closed surface \({\textbf{S}}={\textbf{S}}_\Lambda \), the boundary of a regular neighbourhood of an embedding of \(\Lambda \) 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 \({{\mathcal {M}}}\) of \({\textbf{S}}\) has a symplectic structure and associated Liouville measure; on the other hand, the character variety \({\textbf{M}}\) of \(\Lambda \) carries a natural measure inherited from the Haar measure. Loops on \({\textbf{S}}\) 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 \({{\mathcal {M}}}\). We develop a calculus for calculating correlations of Wilson loops on \({{\mathcal {M}}}\) w.r.to the normalised Liouville measure, and present evidence that they approximate—for large graphs—the corresponding integrals over \({\textbf{M}}\). Lattice field theory involves integrals over \({\textbf{M}}\); we present “symplectic” analogues of expressions for partition functions, Wilson loop expectations, etc., in two and three space-time dimensions.