#### Volume 7, issue 1 (2003)

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The sigma orientation for analytic circle-equivariant elliptic cohomology

### Matthew Ando

Geometry & Topology 7 (2003) 91–153
 arXiv: math.AT/0201092
##### Abstract

We construct a canonical Thom isomorphism in Grojnowski’s equivariant elliptic cohomology, for virtual $\mathbb{T}$–oriented $\mathbb{T}$–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
##### 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