Differential Geometry
Equivariant gerbes over compact simple Lie groups
[Gerbes equivariantes sur les groupes de Lie simples compacts]

Présenté par : Charles-Michel Marle

Behrend, Kai1 ; Xu, Ping2 ; Zhang, Bin3
1 Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver BC, V6T IZ2, Canada
2 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
3 Department of Mathematics, State University of New York at Stony Brook, Stony Brook, NY 11794-3600, USA
Comptes Rendus. Mathématique, Tome 336 (2003) no. 3, pp. 251-256.

En utilisant des extensions S1-centrales de groupoı̈des, nous présentons, dans le cas d'un groupe simple compact G, un modèle de dimension infinie d'une S1-gerbe sur un champ différentiable G/G dont la classe de Dixmier–Douady correspond au générateur canonique de la cohomologie équivariante HG3(G).

Using groupoid S1-central extensions, we present, for a compact simple Lie group G, an infinite dimensional model of S1-gerbe over the differential stack G/G whose Dixmier–Douady class corresponds to the canonical generator of the equivariant cohomology HG3(G).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)00024-9
Behrend, Kai 1 ; Xu, Ping 2 ; Zhang, Bin 3

1 Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver BC, V6T IZ2, Canada
2 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
3 Department of Mathematics, State University of New York at Stony Brook, Stony Brook, NY 11794-3600, USA 
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Behrend, Kai; Xu, Ping; Zhang, Bin. Equivariant gerbes over compact simple Lie groups. Comptes Rendus. Mathématique, Tome 336 (2003) no. 3, pp. 251-256. doi : 10.1016/S1631-073X(02)00024-9. http://www.numdam.org/articles/10.1016/S1631-073X(02)00024-9/

[1] Alekseev, A.; Malkin, A.; Meinrenken, E. Lie group valued moment maps, J. Differential Geom., Volume 48 (1998), pp. 445-495

[2] Behrend, K.; Xu, P. S1-bundles and gerbes over differential stacks, C. R. Acad. Sci. Paris Sér. I, Volume 336 (2003)

[3] K. Behrend, P. Xu, Differential stacks and gerbes, in preparation

[4] Brylinski, J.-L. Gerbes on complex reductive Lie groups | arXiv

[5] Huebschmann, J.; Guruprasad, K.; Jeffrey, L.; Weinstein, A. Group systems, groupoids, and moduli spaces of parabolic bundles, Duke Math. J., Volume 89 (1997), pp. 377-412

[6] Meinrenken, E. The basic gerbe over a compact simple Lie group | arXiv

[7] Pressley, A.; Segal, G. Loop Groups, Oxford University Press, New York, 1986

[8] Weinstein, A. The symplectic structure on moduli space, The Floer Memorial Volume, Progr. Math., 133, 1995, pp. 627-635

[9] Weinstein, A.; Xu, P. Extensions of symplectic groupoids and quantization, J. Reine Angew. Math., Volume 417 (1991), pp. 159-189

[10] Xu, P. Morita equivalent symplectic groupoids (Dazord, P.; Weinstein, A., eds.), Symplectic Geometry, Groupoids, and Integrable Systems, Seminaire sud Rhodanien a Berkeley, 1989, 1991, pp. 291-311

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