Gauge theory and string topology (Q2398332)

The authors study the interplay of the gauge group of a \(G\)-principal bundle \(P\) and its string topology spectrum \(\mathcal S(P)\). The setup is as follows. Let \(M\) be a closed smooth connected based \(n\)-manifold, let \(G\) be a topological group, and let \(P\) be a principal \(G\)-bundle over \(M\). Then one can form the adjoint bundle \(P^{\text{ad}} = P \times_G G\) over \(M\), pull back the bundle \(- TM\) over \(M\) to \(P^{\text{ad}}\), and form the Thom spectrum \(\mathcal S(P) = (P^{\text{ad}})^{-TM}\) associated with this virtual vector bundle. This ``string topology spectrum'' has been studied by \textit{K. Gruher} and \textit{P. Salvatore} [Proc. Lond. Math. Soc. (3) 96, No. 1, 78--106 (2008; Zbl 1143.57012)] who relate the product structure on its homology to the Chas-Sullivan product on the shifted homology of the free loop space on \(M\). When the total space of \(P\) is contractible, \(\mathcal S(P)\) is the Thom spectrum \(LM^{-TM}\) studied in earlier work of \textit{R. L. Cohen} and \textit{J. D. S. Jones} [Math. Ann. 324, No. 4, 773--798 (2002; Zbl 1025.55005)]. In the present paper, the authors examine the string topology spectrum \(\mathcal S(P)\) using the language of \(E\)-line bundles over a ring spectrum \(E\) introduced by \textit{M. Ando} et al. [J. Topol. 7, No. 3, 869--893 (2014; Zbl 1312.55011)]. For example, they use a fiberwise suspension to form a \(\Sigma^{\infty}(G_+)\)-line bundle \(\Sigma^{\infty}_M(P_+)\) and show that the endomorphism ring spectrum of \(\Sigma^{\infty}_M(P_+)\) is equivalent to \(\mathcal S(P)\). The gauge group \(\mathcal G(P)\) of continuous automorphisms of \(P\) over the identity of \(M\) can be identified with a suitable group of sections of \(P^{\text{ad}}\). This description provides a map of grouplike monoids from \(\mathcal G(P)\) to the homotopy automorphism space of \( \Sigma^{\infty}_M(P_+)\) and thus a monoid homomorphism from \(\mathcal G(P)\) to the units of \(\mathcal S(P)\). The authors explore various aspects of this action of \(\mathcal G(P)\) on \(\mathcal S(P)\). For example, they define a category of connections associated with an \(E\)-line bundle over \(M\) and show that its nerve has the homotopy type of the classifying space of the homotopy automorphism space of the \(E\)-line bundle. In the case of \( \Sigma^{\infty}_M(P_+)\), this relates the classifying space of the units of \(\mathcal S(P)\) to connections.