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

Ramras, Daniel A

doi:10.2140/agt.2008.8.2209

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

Daniel A Ramras

Algebraic & Geometric Topology 8 (2008) 2209–2251

DOI: 10.2140/agt.2008.8.2209

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 $B Γ$ . 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 homotopy $K_{*}^{def} (π_{1} Σ) ≅ K^{- *} (Σ)$ for 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.

Keywords

Atiyah–Segal theorem, deformation $K$–theory, flat connection, Yang–Mills theory

Mathematical Subject Classification 2000

Primary: 55N15, 58E15

Secondary: 58D27, 19L41

References

Publication

Received: 14 May 2008

Revised: 17 October 2008

Accepted: 26 October 2008

Published: 4 December 2008

Authors

Daniel A Ramras
Vanderbilt University Department of Mathematics 1326 Stevenson Center Nashville, TN 37240 USA
http://www.math.vanderbilt.edu/~ramrasda

Volume 8, issue 4 (2008)