A model for complex analytic equivariant elliptic cohomology from quantum field theory

TL;DR

This paper develops a geometric model for complex analytic equivariant elliptic cohomology using quantum field theory, linking physical theories with advanced mathematical structures like elliptic curves and cohomology classes.

Contribution

It introduces a global geometric construction of equivariant elliptic cohomology for all compact Lie groups via quantum field theory, connecting physical models with elliptic cohomology concepts.

Findings

Constructs cocycles from 2D sigma model fields with supersymmetry.

Identifies fermion partition functions with equivariant elliptic Euler classes.

Shows moduli stack of U(1)-gauge fields has a multiplication compatible with elliptic curve structure.

Abstract

We construct a global geometric model for complex analytic equivariant elliptic cohomology for all compact Lie groups. Cocycles are specified by functions on the space of fields of the two-dimensional sigma model with background gauge fields and $\mathcal{N} = (0, 1)$ supersymmetry. We also consider a theory of free fermions valued in a representation whose partition function is a section of a determinant line bundle. We identify this section with a cocycle representative of the (twisted) equivariant elliptic Euler class of the representation. Finally, we show that the moduli stack of $U(1)$-gauge fields carries a multiplication compatible with the complex analytic group structure on the universal (dual) elliptic curve, with the Euler class providing a choice of coordinate. This provides a physical manifestation of the elliptic group law central to the homotopy-theoretic construction of elliptic cohomology.