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(Γ) of a compact Lie group Γ to the complex K -theory of the classifying space BΓ. 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 (Γ) (the homotopy-theoretical analogue of R(Γ)). Our main theorem provides an isomorphism in homotopyfor all compact, aspherical surfaces Σ and all * > 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.

The stable moduli space of flat connections over a surface

“…By the Dold-Thom Theorem, the latter groups are precisely the reduced integral homology groups of M g . For i = 1, 2, the groups π i R def (M g ) were computed in Ramras [39,Section 6] and found to agree with the integral homology groups of M g ; these also agree with π * (M flat (M g )) by Lemma 5.3. To complete the proof, we must show that for * > 2 the groups π…”

“…We begin by briefly reviewing deformation K-theory. For further details and discussion, see [25,35,39]. and it may be thought of as the homotopical analogue of the representation ring R(Γ) (the ordinary group completion of the discrete monoid of isomorphism classes of representations).…”

“…For the surface groups studied in this paper, Rep(−) is stably group-like with respect to the trivial representation 1 ∈ Hom(−, U (1)) (see Ramras [39,Section 4] or Ho and Liu [19]), and we simply say that Rep(−) is stably group-like.…”

“…We briefly review the notion of a flat connection and its holonomy. For further details on these concepts and other gauge-theoretical material below, we refer the reader to Ramras [39] or Wehrheim [44].…”

“…(We recall the construction of φ A below; see Ramras [39,Appendix] for full details.) Note that the mixing construction can also be applied to the map Eπ 1 S → Bπ 1 S, resulting in the bundle…”