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LOOP GROUPS, STRING CLASSES AND EQUIVARIANT COHOMOLOGY

  • RAYMOND F. VOZZO (a1)

Abstract

We give a classifying theory for LG-bundles, where LG is the loop group of a compact Lie group G, and present a calculation for the string class of the universal LG-bundle. We show that this class is in fact an equivariant cohomology class and give an equivariant differential form representing it. We then use the caloron correspondence to define (higher) characteristic classes for LG-bundles and to prove a result for characteristic classes for based loop groups for the free loop group. These classes have a natural interpretation in equivariant cohomology and we give equivariant differential form representatives for the universal case in all odd dimensions.

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References

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[1]Alekseev, A. and Meinrenken, E., ‘The Atiyah algebroid of the path fibration over a Lie group’, Lett. Math. Phys. 90 (2009), 2358.
[2]Atiyah, M. F. and Bott, R., ‘The moment map and equivariant cohomology’, Topology 23(1) (1984), 128.
[3]Carey, A. L. and Mickelsson, J., ‘The universal gerbe, Dixmier–Douady class, and gauge theory’, Lett. Math. Phys. 59(1) (2002), 4760.
[4]Cartan, H., ‘Notions d’algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie’, in: Colloque de topologie (espaces fibrés), Bruxelles, 1950 (Georges Thone, Liège, 1951), pp. 1527.
[5]Dupont, J. L., Curvature and Characteristic Classes, Lecture Notes in Mathematics, 640 (Springer, Berlin, 1978).
[6]Garland, H. and Murray, M. K., ‘Kac–Moody monopoles and periodic instantons’, Comm. Math. Phys. 120(2) (1988), 335351.
[7]Guillemin, V. W. and Sternberg, S., Supersymmetry and Equivariant de Rham Theory, Mathematics Past and Present (Springer, Berlin, 1999), With an appendix containing two reprints by Henri Cartan.
[8]Jeffrey, L. C., ‘Group cohomology construction of the cohomology of moduli spaces of flat connections on 2-manifolds’, Duke Math. J. 77(2) (1995), 407429.
[9]Kalkman, J., ‘A BRST model applied to symplectic geometry’, PhD Thesis, Universiteit Utrecht, 1993.
[10]Killingback, T. P., ‘World-sheet anomalies and loop geometry’, Nuclear Phys. B 288(3–4) (1987), 578588.
[11]Mathai, V. and Quillen, D., ‘Superconnections, Thom classes, and equivariant differential forms’, Topology 25(1) (1986), 85110.
[12]Murray, M. K. and Stevenson, D., ‘Higgs fields, bundle gerbes and string structures’, Comm. Math. Phys. 243(3) (2003), 541555.
[13]Murray, M. K. and Vozzo, R. F., ‘The caloron correspondence and higher string classes for loop groups’, J. Geom. Phys. 60(9) (2010), 12351250.
[14]Murray, M. K. and Vozzo, R. F., ‘Circle actions, central extensions and string structures’, Int. J. Geom. Methods Mod. Phys. 7(6) (2010), 10651092.
[15]Narasimhan, M. S. and Ramanan, S., ‘Existence of universal connections’, Amer. J. Math. 83 (1961), 563572.
[16]Narasimhan, M. S. and Ramanan, S., ‘Existence of universal connections. II’, Amer. J. Math. 85 (1963), 223231.
[17]Pressley, A. and Segal, G., Loop Groups, Oxford Mathematical Monographs (Oxford University Press, New York, 1986).
[18]Schlafly, R., ‘Universal connections’, Invent. Math. 59(1) (1980), 5965.
[19]Vozzo, R. F., ‘Loop groups, Higgs fields and generalised string classes’, PhD Thesis, School of Mathematical Sciences, University of Adelaide, 2009.
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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