Yang–Mills theory over surfaces and the Atiyah–Segal theorem

Abstract

In this paper we explain how Morse theory for the Yang–Mills functional can be used to prove an analogue for surface groups of the Atiyah–Segal theorem. Classically, the Atiyah–Segal theorem relates the representation ring $R\left(\Gamma \right)$ of a compact Lie group $\Gamma$ to the complex $K$–theory of the classifying space $B\Gamma$. For infinite discrete groups, it is necessary to take into account deformations of representations, and with this in mind we replace the representation ring by Carlsson’s deformation $K$–theory spectrum $K^{def}\left(\Gamma \right)$ (the homotopy-theoretical analogue of $R\left(\Gamma \right)$). Our main theorem provides an isomorphism in homotopy $K_{\ast }^{def}\left({\pi }_1\Sigma \right)\cong K^{-\ast }\left(\Sigma \right)$ for all compact, aspherical surfaces $\Sigma$ and all $\ast >0$. Combining this result with work of Tyler Lawson, we obtain homotopy theoretical information about the stable moduli space of flat unitary connections over surfaces.