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

Ramras, Daniel A¹

¹Vanderbilt University, Department of Mathematics, 1326 Stevenson Center, Nashville, TN 37240, USA

Algebraic and Geometric Topology, Tome 8 (2008) no. 4, pp. 2209-2251

Résumé

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.

Détail

DOI : 10.2140/agt.2008.8.2209

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

Bibliographie

[1] R Abraham, Differential topology, Lecture notes of Smale (1964)

[2] R Abraham, J Robbin, Transversal mappings and flows, W. A. Benjamin (1967)

[3] A Adem, Characters and $K$–theory of discrete groups, Invent. Math. 114 (1993) 489

[4] A Alekseev, A Malkin, E Meinrenken, Lie group valued moment maps, J. Differential Geom. 48 (1998) 445

[5] M F Atiyah, Characters and cohomology of finite groups, Inst. Hautes Études Sci. Publ. Math. (1961) 23

[6] M F Atiyah, R Bott, The Yang–Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983) 523

[7] M F Atiyah, G B Segal, Equivariant $K$–theory and completion, J. Differential Geometry 3 (1969) 1

[8] G Baumslag, E Dyer, A Heller, The topology of discrete groups, J. Pure Appl. Algebra 16 (1980) 1

[9] G Carlsson, Derived representation theory and the algebraic $K$–theory of fields

[10] G Carlsson, Derived completions in stable homotopy theory, J. Pure Appl. Algebra 212 (2008) 550

[11] G D Daskalopoulos, The topology of the space of stable bundles on a compact Riemann surface, J. Differential Geom. 36 (1992) 699

[12] G D Daskalopoulos, K K Uhlenbeck, An application of transversality to the topology of the moduli space of stable bundles, Topology 34 (1995) 203

[13] S K Donaldson, P B Kronheimer, The geometry of four-manifolds, Oxford Math. Monogr., Oxford Science Publ., The Clarendon Press, Oxford University Press (1990)

[14] A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, Math. Surveys and Monogr. 47, Amer. Math. Soc. (1997)

[15] A D Elmendorf, M A Mandell, Rings, modules, and algebras in infinite loop space theory, Adv. Math. 205 (2006) 163

[16] B Fine, P Kirk, E Klassen, A local analytic splitting of the holonomy map on flat connections, Math. Ann. 299 (1994) 171

[17] P Griffiths, J Harris, Principles of algebraic geometry, Pure and Applied Math., Wiley-Interscience (1978)

[18] K Gruher, String topology of classifying spaces, PhD thesis, Stanford University (2007)

[19] G Harder, M S Narasimhan, On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann. 212 (1975) 215

[20] A Hatcher, Algebraic topology, Cambridge University Press (2002)

[21] G Higman, A finitely generated infinite simple group, J. London Math. Soc. 26 (1951) 61

[22] H Hironaka, Triangulations of algebraic sets, from: "Algebraic geometry (Humboldt State Univ., Arcata, Calif., 1974)", Proc. Sympos. Pure Math. 29, Amer. Math. Soc. (1975) 165

[23] N K Ho, The real locus of an involution map on the moduli space of flat connections on a Riemann surface, Int. Math. Res. Not. (2004) 3263

[24] N K Ho, C C M Liu, Connected components of the space of surface group representations, Int. Math. Res. Not. (2003) 2359

[25] N K Ho, C C M Liu, On the connectedness of moduli spaces of flat connections over compact surfaces, Canad. J. Math. 56 (2004) 1228

[26] N K Ho, C C M Liu, Connected components of spaces of surface group representations. II, Int. Math. Res. Not. (2005) 959

[27] N K Ho, C C M Liu, Yang–Mills connections on nonorientable surfaces, Comm. Anal. Geom. 16 (2008) 617

[28] M Hovey, B Shipley, J Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149

[29] J Kurzweil, On approximation in real Banach spaces, Studia Math. 14 (1955) 214

[30] S Lang, Differential and Riemannian manifolds, Graduate Texts in Math. 160, Springer (1995)

[31] T Lawson, Derived representation theory of nilpotent groups, PhD thesis, Stanford University (2004)

[32] T Lawson, The product formula in unitary deformation $K$–theory, $K$–Theory 37 (2006) 395

[33] T Lawson, The Bott cofiber sequence in deformation $K$–theory and simultaneous similarity in $U(n)$, to appear in Math. Proc. Cambridge Philos. Soc. (2008)

[34] W Lück, Rational computations of the topological $K$–theory of classifying spaces of discrete groups, J. Reine Angew. Math. 611 (2007) 163

[35] W Lück, B Oliver, The completion theorem in $K$–theory for proper actions of a discrete group, Topology 40 (2001) 585

[36] D Mcduff, G Segal, Homology fibrations and the “group-completion” theorem, Invent. Math. 31 (1976) 279

[37] J Milnor, Construction of universal bundles. II, Ann. of Math. $(2)$ 63 (1956) 430

[38] P K Mitter, C M Viallet, On the bundle of connections and the gauge orbit manifold in Yang–Mills theory, Comm. Math. Phys. 79 (1981) 457

[39] S Morita, Geometry of characteristic classes, Transl. of Math. Monogr. 199, Amer. Math. Soc. (2001)

[40] S Morita, Geometry of differential forms, Trans. of Math. Monogr. 201, Amer. Math. Soc. (2001)

[41] D H Park, D Y Suh, Linear embeddings of semialgebraic $G$–spaces, Math. Z. 242 (2002) 725

[42] J Råde, On the Yang–Mills heat equation in two and three dimensions, J. Reine Angew. Math. 431 (1992) 123

[43] D Ramras, The Yang–Mills stratification for surfaces revisited

[44] D A Ramras, Excision for deformation $K$–theory of free products, Algebr. Geom. Topol. 7 (2007) 2239

[45] D Ramras, Stable representation theory of infinite discrete groups, PhD thesis, Stanford University (2007)

[46] G Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. (1968) 105

[47] G Segal, Categories and cohomology theories, Topology 13 (1974) 293

[48] M R Sepanski, Compact Lie groups, Graduate Texts in Math. 235, Springer (2007)

[49] M Thaddeus, An introduction to the topology of the moduli space of stable bundles on a Riemann surface, from: "Geometry and physics (Aarhus, 1995)" (editors J E Andersen, J Dupont, H Pedersen, A Swann), Lecture Notes in Pure and Appl. Math. 184, Dekker (1997) 71

[50] K K Uhlenbeck, Connections with $L^{p}$ bounds on curvature, Comm. Math. Phys. 83 (1982) 31

[51] K Wehrheim, Uhlenbeck compactness, EMS Series of Lectures in Math., Eur. Math. Soc. (2004)

[52] H Whitney, Elementary structure of real algebraic varieties, Ann. of Math. $(2)$ 66 (1957) 545

[53] A Wilansky, Topology for analysis, Robert E. Krieger Publishing Co. (1983)