Gauge theory and string topology

Abstract

Given a principal bundle over a closed manifold, G --> P --> M, let P^{Ad} --> M be the associated adjoint bundle. Gruher and Salvatore showed that the Thom spectrum (P^{Ad})^{-TM} is a ring spectrum whose corresponding product in homology is a Chas-Sullivan type string topology product. We refer to this spectrum as the `string topology spectrum of P", S (P). In the universal case when P is contractible, S(P) is equivalent to LM^{-TM} where LM is the free loop space of the manifold. This ring spectrum was introduced by the authors as a homotopy theoretic realization of the Chas-Sullivan string topology of M. The main purpose of this paper is to introduce an action of the gauge group of the principal bundle, G (P) on the string topology spectrum S(P), and to study this action in detail. Indeed we study the entire group of units and the induced representation G(P) --> GL_1(S (P)). We show that this group of units is the group of homotopy automorphisms of the fiberwise suspension spectrum of P. More generally we describe the homotopy type of the group of homotopy automorphisms of any E-line bundle for any ring spectrum E. We import some of the basic ideas of gauge theory, such as the action of the gauge group on the space of connections to the setting of E-line bundles over a manifold, and do explicit calculations. We end by discussing a functorial perspective, which describes a sense in which the string topology spectrum S(P) of a principal bundle is the "linearization" of the gauge group G(P).