Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 8, Number 4 (2008), 2209-2251.
Yang–Mills theory over surfaces and the Atiyah–Segal theorem
Full-text: Open access
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 of a compact Lie group to the complex –theory of the classifying space . 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 –theory spectrum (the homotopy-theoretical analogue of ). Our main theorem provides an isomorphism in homotopy for all compact, aspherical surfaces and all . Combining this result with work of Tyler Lawson, we obtain homotopy theoretical information about the stable moduli space of flat unitary connections over surfaces.
Article information
Source
Algebr. Geom. Topol., Volume 8, Number 4 (2008), 2209-2251.
Dates
Received: 14 May 2008
Revised: 17 October 2008
Accepted: 26 October 2008
First available in Project Euclid: 20 December 2017
Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796932
Digital Object Identifier
doi:10.2140/agt.2008.8.2209
Mathematical Reviews number (MathSciNet)
MR2465739
Zentralblatt MATH identifier
1240.19005
Subjects
Primary: 55N15: $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19- XX} 58E15: Application to extremal problems in several variables; Yang-Mills functionals [See also 81T13], etc.
Secondary: 58D27: Moduli problems for differential geometric structures 19L41: Connective $K$-theory, cobordism [See also 55N22]
Keywords
Atiyah–Segal theorem deformation $K$–theory flat connection Yang–Mills theory
Citation
Ramras, Daniel A. Yang–Mills theory over surfaces and the Atiyah–Segal theorem. Algebr. Geom. Topol. 8 (2008), no. 4, 2209--2251. doi:10.2140/agt.2008.8.2209. https://projecteuclid.org/euclid.agt/1513796932
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