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