Physical phase space of lattice Yang-Mills theory and the moduli space of flat connections on a Riemann surface

TL;DR

This paper demonstrates that the physical phase space of a lattice Yang-Mills theory with a deformation parameter is equivalent to the moduli space of flat connections on a Riemann surface, linking it to a 3D Chern-Simons model.

Contribution

It establishes a precise Poisson isomorphism between the deformed lattice Yang-Mills phase space and the moduli space of flat connections, connecting lattice gauge theory with geometric structures.

Findings

Phase space coincides with moduli space of flat connections

Deformation parameter relates to Chern-Simons level

Links and vertices determine the genus of the surface

Abstract

It is shown that the physical phase space of $\g$-deformed Hamiltonian lattice Yang-Mills theory, which was recently proposed in refs.[1,2], coincides as a Poisson manifold with the moduli space of flat connections on a Riemann surface with $(L-V+1)$ handles and therefore with the physical phase space of the corresponding $(2+1)$-dimensional Chern-Simons model, where $L$ and $V$ are correspondingly a total number of links and vertices of the lattice. The deformation parameter $\g$ is identified with $\frac {2\pi}{k}$ and $k$ is an integer entering the Chern-Simons action.