Kan group and lattice gauge theory

  • Johannes Huebschmann (Université des Sciences et Technologies de Lille, U. F. R. de Mathématiques, France)
A3 01 (Sophus-Lie room)


Let $Y$ be a cell complex with a single 0-cell, let $K$ be its Kan group, a free simplicial group whose combinatorial structure reflects the incidence structure of $Y$ and which is a model for the based loop space $\Omega Y$ of $Y$, and let $G$ be a Lie group. We will describe the construction of a weak $G$-equivariant homotopy equivalence from the geometric realization $|\textnormal{Hom}(K,G)|$ of the cosimplicial manifold $\textnormal{Hom}(K,G)$ of homomorphisms from $K$ to $G$ to the space $\textnormal{Map}^o(Y,BG)$ of based maps from $Y$ to the classifying space $BG$ of $G$ where $G$ acts on $BG$ by conjugation. In this fashion, $|\textnormal{Hom}(K,G)|$ appears as a model for the space of based gauge equivalence classes of connections. Combined with an explicit purely finite dimensional construction of generators of the equivariant cohomology of the geometric realization of $\textnormal{Hom}(K,G)$ and hence of $\textnormal{Map}^o(Y,BG)$, this construction, when carried out for the special case where $Y$ underlies a smooth manifold, may be viewed as a rigorous approach to lattice gauge theory. Under these circumstances it yields, (i) when {$\textnormal{dim}(Y)=2$,} equivariant de Rham representatives of generators of the equivariant cohomology of certain moduli spaces and (ii) when {$\textnormal{dim}(Y)=3$,} equivariant cohomology generators including a rigorous combinatorial description of the Chern-Simons function for a closed 3-manifold.

More details may be found in:
J. Huebschmann: Extended moduli spaces, the Kan construction, and lattice gauge theory; Topology, 38, 1999, 555--596

The Kan group may be viewed as a combinatorial version of the hoop group\/} introduced by Ashtekar et al, and $|\textnormal{Hom}(K,G)|$ is a combinatorial version of the space of homomorphims from the hoop group to the structure group explored by Ashtekar et al.