Extended moduli spaces and the Kan construction.II.Lattice gauge theory

TL;DR

This paper provides a finite-dimensional, explicit construction of generators for the equivariant cohomology of certain mapping spaces, offering a rigorous approach to lattice gauge theory with applications in geometry and topology across dimensions 2, 3, and 4.

Contribution

It introduces a novel finite-dimensional method to generate equivariant cohomology for mapping spaces, connecting lattice gauge theory with geometric and topological invariants.

Findings

Generators for equivariant cohomology in dimension 2 include those for moduli spaces of vector bundles.

In dimension 3, generators include the Chern-Simons function.

In dimension 4, generators relate to Donaldson polynomials.

Abstract

Let $Y$ be a CW-complex with a single 0-cell, $K$ its Kan group, a model for the loop space of $Y$, and let $G$ be a compact, connected Lie group. We give an explicit finite dimensional construction of generators of the equivariant cohomology of the geometric realization of the cosimplicial manifold $\roman{Hom}(K,G)$ and hence of the space $\roman{Map}^o(Y,BG)$ of based maps from $Y$ to the classifying space $BG$. For a smooth manifold $Y$, this may be viewed as a rigorous approach to lattice gauge theory, and we show that it then yields, (i) when {$\roman{dim}(Y)=2$,} equivariant de Rham representatives of generators of the equivariant cohomology of twisted representation spaces of the fundamental group of a closed surface including generators for moduli spaces of semi stable holomorphic vector bundles on complex curves so that, in particular, the known structure of a stratified symplectic space results; (ii) when {$\roman{dim}(Y)=3$,} equivariant cohomology generators including the Chern-Simons function; (iii) when {$\roman{dim}(Y) = 4$,} the generators of the relevant equivariant cohomology from which for example Donaldson polynomials are obtained by evaluation against suitable fundamental classes corresponding to moduli spaces of ASD connections.