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

TL;DR

This paper explores the Chern character of infinite rank bundles in gauge theory, using cocycles to define characteristic classes even when traditional methods fail due to infinite dimensionality.

Contribution

It introduces a method to define the Chern character for infinite dimensional bundles via cocycles with trace-class representations, applicable in gauge theory contexts.

Findings

Chern character can be defined for certain infinite rank bundles using cocycles.

The method applies to gauge connection spaces and gauge transformation groups.

Connections to quantum field theory inform the construction of cocycles.

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.