Chern-Simons theory and bundles of infinite-dimensional matrix algebras

Jean-Luc Brylinski

Abstract: For a manifold $X$, all classes in $H^3(X,Z)$ arise from bundles of matrix algebras (of infinite size). For $G=SL(n,C)$, the generator of $H^3(G,Z)=Z$ corersponds to the well-known Chern-Simons form. Motivated by loop groups and field theory, I would like to construct an explicit bundle of matrix algebras in this cohomology class. So far I can give an explicit construction for such a bundle over the regular semisimple locus of $G$. Moreover I can extend the stable equivalence class of this bundle to the whole group $G$; this gives an equivariant gerbe over $G$. The problem of finding an explicit global bundle remains open.

created Mon Dec 1 13:46:10 PST 2003