The sigma orientation for analytic circle-equivariant elliptic cohomology

We construct a canonical Thom isomorphism in Grojnowski’s equivariant elliptic cohomology, for virtual $T$ –oriented $T$ –equivariant spin bundles with vanishing Borel-equivariant second Chern class, which is natural under pull-back of vector bundles and exponential under Whitney sum. It extends in the complex-analytic case the non-equivariant sigma orientation of Hopkins, Strickland, and the author. The construction relates the sigma orientation to the representation theory of loop groups and Looijenga’s weighted projective space, and sheds light even on the non-equivariant case. Rigidity theorems of Witten-Bott-Taubes including generalizations by Kefeng Liu follow.

Keywords

Sigma orientation, equivariant elliptic cohomolgy, rigidity

Mathematical Subject Classification 2000

Primary: 55N34

Secondary: 55N22, 57R91

References

Forward citations

Publication

Received: 1 February 2002

Revised: 18 October 2003

Accepted: 19 November 2002

Published: 17 February 2003

Proposed: Haynes Miller

Seconded: Ralph Cohen, Gunnar Carlsson

Authors

Matthew Ando
Department of Mathematics University of Illinois at Urbana-Champaign Urbana Illinois 61801 USA

Volume 7, issue 1 (2003)

Matthew Ando