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.

Katharina Matschke

MPI for Mathematics in the Sciences Contact via Mail