Gauge theory and string topology

Abstract

Given a principal bundle over a closed manifold, $G \rightarrow P \rightarrow M$, let $P^{Ad} \rightarrow M$ be the associated adjoint bundle. Gruher and Salvatore (Proc Lond Math Soc 96(3), 78106 2008) showed that the Thom spectrum $(P^\mathrm{Ad})^{-TM}$ 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”, $\mathcal {S}(P)$. In the universal case when P is contractible, $\mathcal {S}(P) \simeq LM^{-TM}$ where LM is the free loop space of the manifold. This ring spectrum was introduced by the authors in Cohen et al. (Math Annalen 324, 773–798 2002) 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, $\mathcal {G}(P)$ on the string topology spectrum $\mathcal {S}(P)$, and to study this action in detail. Indeed we study the entire group of units and the induced representation $\mathcal {G}(P) \rightarrow GL_1(\mathcal {S}(P))$. 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 $\mathcal {S}(P)$ of a principal bundle is the “linearization” of the gauge group $\mathcal {G}(P)$.