Electronic Research Announcements

Intersection pairings in moduli spaces of holomorphic bundles on a Riemann surface

Author(s): Lisa C. Jeffrey; Frances C. Kirwan
Journal: Electron. Res. Announc. Amer. Math. Soc. 1 (1995), 57-71.
MSC (1991): Primary 58F05, 14F05, 53C05
Retrieve article in: PDF

Abstract: We outline a proof of formulas (found by Witten in 1992 using physical methods) for intersection pairings in the cohomology of the moduli space $M(n,d)$

of stable holomorphic vector bundles of rank $n$

and degree $d$

(assumed coprime) and fixed determinant on a Riemann surface of genus $g \ge 2$ .

1: M.F. Atiyah, R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. Lond. A308 (1982) 523-615. MR 85k:14006
2: M.F. Atiyah, R. Bott, The moment map and equivariant cohomology, Topology 23 (1984) 1-28. MR 85e:58041
3: N. Berline, E. Getzler, M. Vergne, Heat Kernels and Dirac Operators, Springer-Verlag (Grundlehren vol. 298), 1992. MR 94e:58130
4: N. Berline, M. Vergne, Classes caractéristiques équivariantes. Formules de localisation en cohomologie équivariante, C. R. Acad. Sci. Paris 295 (1982) 539-541. MR 83m:58002
5: N. Berline, M. Vergne, Zéros d'un champ de vecteurs et classes caractéristiques équivariantes, Duke Math. J. 50 (1983) 539-549. MR 84i:58114
6: T. Bröcker, T. tom Dieck, Representations of Compact Lie Groups (Graduate Texts in Mathematics vol. 98) Springer-Verlag, 1985. MR 86i:22023
7: S.K. Donaldson, Gluing techniques in the cohomology of moduli spaces, in Topological Methods in Modern Mathematics (Proceedings of 1991 conference in Stony Brook, NY in honour of the sixtieth birthday of J. Milnor), Publish or Perish. MR 94b:57036
8: J.J Duistermaat, G. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982) 259-268; MR 84h:58051a; Addendum, 72 (1983) 153-158. MR 84h:58051b
9: J.J. Duistermaat, Equivariant cohomology and stationary phase, Utrecht preprint no. 817 (1993).
10: M. Duflo, M. Vergne, Orbites coadjointes et cohomologie équivariante, in M. Duflo, N.V. Pedersen, M. Vergne (ed.), The Orbit Method in Representation Theory (Progress in Mathematics, vol. 82), Birkhäuser, (1990) 11-60. MR 93b:22013
11: V. Guillemin, J. Kalkman, A new proof of the Jeffrey-Kirwan localization theorem, to appear (1994).
12: S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, 1978. MR 80k:53081
13: L. Hörmander, The Analysis of Linear Partial Differential Operators I (Grundlehren v. 256), Springer, 1983. MR 85g:35002a
14: L.C. Jeffrey, Extended moduli spaces of flat connections on Riemann surfaces, Math. Annalen 298 (1994) 667-692.MR 95g:58030
15: L.C. Jeffrey, Symplectic forms on moduli spaces of flat connections on 2-manifolds, to appear in Proceedings of the Georgia International Topology Conference (Athens, GA, 1993), ed. W. Kazez.
16: L.C. Jeffrey, Group cohomology construction of the cohomology of moduli spaces of flat connections on 2-manifolds, Duke Math. J. 77 (1995) 407-429.
17: L.C. Jeffrey, F.C. Kirwan, Localization for nonabelian group actions, Topology 34 (1995) 291-327. CMP 95:08
18: L.C. Jeffrey, F.C. Kirwan, Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface, in preparation.
19: F. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry, Princeton University Press (1984). MR 86i:58050
20: F. Kirwan, The cohomology rings of moduli spaces of vector bundles over Riemann surfaces, J. Amer. Math. Soc., 5 (1992) 853-906. MR 93g:14016
21: S.K. Martin, Cohomology rings of symplectic quotients, preprint (1994).
22: M.S. Narasimhan, C.S. Seshadri, Stable and unitary bundles on a compact Riemann surface, Ann. Math. 82 (1965) 540-567. MR 32:1725
23: A. Szenes, The combinatorics of the Verlinde formula, preprint alg-geom/9402003; A. Szenes, private communication.
24: M. Thaddeus, Conformal field theory and the cohomology of the moduli space of stable bundles, J. Diff. Geom. 35 (1992) 131-149. MR 93g:14017
25: E. Witten, On quantum gauge theories in two dimensions, Commun. Math. Phys. 141 (1991) 153-209. MR 93i:58164
26: E. Witten, Two dimensional gauge theories revisited, preprint hep-th/9204083; J. Geom. Phys. 9 (1992) 303-368. MR 93m:58017

Retrieve articles in Electronic Research Announcements with MSC (1991): 58F05, 14F05, 53C05

Lisa C. Jeffrey
Affiliation: Lisa C. Jeffrey, Mathematics Department, Princeton University, Princeton, NJ 08544, USA
Email: jeffrey@math.princeton.edu

Frances C. Kirwan
Affiliation: Frances C. Kirwan, Balliol College, Oxford OX1 3BJ, UK
Email: fkirwan@vax.ox.ac.uk

DOI: 10.1090/S1079-6762-95-02002-6
PII: S 1079-6762(95)02002-6
Keywords: Moduli spaces, symplectic geometry, intersection pairings
Received by editor(s): June 28, 1995
Additional Notes: This material is based on work supported by the National Science Foundation under Grant. No. DMS-9306029.
Copyright of article: Copyright 1995, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1079-6762

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article

Currently published by the American Institute of Mathematical Sciences as Electronic Research Announcements in Mathematical Sciences


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS