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Yang–Mills theory over surfaces and the
Atiyah–Segal theorem
Daniel A Ramras
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Algebraic & Geometric Topology 8
(2008) 2209–2251
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Abstract
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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 Kdef(Γ) (the homotopy-theoretical
analogue of R(Γ)). Our main theorem provides an isomorphism in homotopy
K*def(π1Σ)≅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.
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Keywords
Atiyah–Segal theorem, deformation
K–theory, flat connection, Yang–Mills theory
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Mathematical Subject Classification
Primary: 55N15, 58E15
Secondary: 19L41, 58D27
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Publication
Received: 14 May 2008
Revised: 17 October 2008
Accepted: 26 October 2008
Published: 4 December 2008
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