A tower connecting gauge groups to string topology

TL;DR

This paper develops a calculus of functors to relate gauge groups of principal bundles to Thom ring spectra in string topology, extending known linear approximations to higher orders and providing explicit calculations.

Contribution

It introduces higher-order approximations of the gauge group via calculus of functors, generalizing previous linear models and explicit computations in string topology.

Findings

Established higher-order approximations of gauge groups

Explicitly calculated these approximations

Extended previous work by Cohen, Jones, and Arone

Abstract

We develop a variant of calculus of functors, and use it to relate the gauge group G(P) of a principal bundle P over M to the Thom ring spectrum (P^Ad)^{-TM}. If P has contractible total space, the resulting Thom ring spectrum is LM^{-TM}, which plays a central role in string topology. R.L. Cohen and J.D.S. Jones have recently observed that, in a certain sense, (P^Ad)^{-TM} is the linear approximation of G(P). We prove an extension of that relationship by demonstrating the existence of higher-order approximations and calculating them explicitly. This also generalizes calculations done by G. Arone.