EXTENDED MODULI SPACES, THE KAN CONSTRUCTION, AND LATTICE GAUGE THEORY

Let Y be a CW-complex with asingle 0-cell, let K be its Kan group, a free simplicial group whose geometric realization is a model for the space ΩY of based loops on Y, and let G be a Lie group. By means of simplicial and cosimplicial techniques involving fundamental results of Kan’s and the standard W- and bar constructions, we obtain a weak G-equivariant homotopy equivalence from the geometric realization |Hom(K, G)| of the cosimplicial manifold Hom(K, G) of homomorphisms from K to G to the space Mapo(Y, BG) of based maps from Y to the classifying space BG of G where G acts on BG by conjugation. Thereafter we carry out an explicit purely finite dimensional construction of generators of the equivariant cohomology of the geometric realization of Hom(K, G) and hence of the space Mapo(Y, BG) of based maps from Y to the classifying space BG of G. 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 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, when G is compact, the known structure of stratified symplectic space on the twisted representation spaces results and (ii) when dim(Y)=3, equivariant cohomology generators including a rigorous combinatorial description of the Chern–Simons function for a closed 3-manifold. The latter is illustrated by a calculation of the Chern–Simons invariants for flat SU(2)-connections over 3-dimensional lens spaces.