Circle-equivariant classifying spaces and the rational equivariant sigma genus

TL;DR

This paper constructs a rational equivariant sigma orientation map between circle-equivariant cobordism and elliptic cohomology spectra, supported by characteristic classes and algebraic geometry, confirming a prior conjecture.

Contribution

It introduces a new map of ring T-spectra for complex elliptic curves, extending the sigma orientation to the equivariant setting and providing a theoretical framework for calculations.

Findings

Construction of the equivariant sigma orientation map

Development of characteristic classes for calculations

Proof of a conjecture by the first author

Abstract

The circle-equivariant spectrum MString_C is the equivariant analogue of the cobordism spectrum MU<6> of stably almost complex manifolds with c_1=c_2=0. Given a rational elliptic curve C, the second author has defined a ring T-spectrum EC representing the associated T-equivariant elliptic cohomology. The core of the present paper is the construction, when C is a complex elliptic curve, of a map of ring T-spectra MString_C --> EC which is the rational equivariant analogue of the sigma orientation of Ando-Hopkins-Strickland. We support this by a theory of characteristic classes for calculation, and a conceptual description in terms of algebraic geometry. In particular, we prove a conjecture of the first author.