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

Given a finite connected graph $\mathrm{\Lambda }$, the space of $SU(2)$ lattice gauge-fields on $\mathrm{\Lambda }$, 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 $\mathrm{\Lambda }$, 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 $\mathrm{\Lambda }$ 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.