The Chern Character of Certain Infinite Rank Bundles arising in Gauge Theory

Abstract

A cocycle $\Omega: P \times G \to H$ taking values in a Lie group $H$ for a free right action of $G$ on $P$ defines a principal bundle $Q$ with the structure group $H$ over $P/G.$ The Chern character of a vector bundle associated to $Q$ defines then characteristic classes on $X.$ This observation becomes useful in the case of infinite dimensional groups. It typically happens that a representation of $G$ is not given by linear operators which differ from the indentity by a trace-class operator. For this reason the Chern character of a vector bundle associated to the principal fibration $P \to P/G$ is ill-defined. But it may happen that the Lie algebra representations of the group $H$ are given in terms of trace-class operators and therefore the Chern character is well-defined; this observation is useful especially if the map $g\mapsto \Omega(p;g)$ is a homotopy equivalence on the image for any $p\in P.$ We apply this method to the case $P= \Cal A,$ the space of gauge connections in a finite-dimensional vector bundle, and $G= \Cal G$ is the group of (based) gauge transformations. The method for constructing the appropriate cocycle $\Omega$ comes from ideas in quantum field theory, used to define the renormalized gauge currents in a Fock space.