Equivariant elliptic cohomology, gauged sigma models, and discrete torsion

TL;DR

This paper connects supersymmetric sigma models and mechanics to equivariant elliptic cohomology and K-theory, revealing how gauge theories produce cocycles, wrong-way maps, and modular invariants, with applications to discrete torsion and string structures.

Contribution

It establishes a novel link between supersymmetric field theories and equivariant elliptic cohomology, including explicit constructions of cocycles and invariants with applications to discrete torsion.

Findings

Functions on fields determine cocycles for equivariant elliptic cohomology.

Gauge theory path integrals produce wrong-way maps and character formulas.

Equivariant Euler classes lead to modular form-valued invariants depending on string structures.

Abstract

For $G$ a finite group, we show that functions on fields for the 2-dimensional supersymmetric sigma model with background $G$-symmetry determine cocycles for complex analytic $G$-equivariant elliptic cohomology. Similar structures in supersymmetric mechanics determine cocycles for equivariant K-theory with complex coefficients. The path integral for gauge theory with a finite group constructs wrong-way maps associated to group homomorphisms. When applied to an inclusion of groups, we obtain the induced character formula of Hopkins, Kuhn, and Ravenel. For the homomorphism $G\to *$ we obtain Vafa's formula for gauging with discrete torsion. The image of equivariant Euler classes under gauging constructs modular form-valued invariants of representations that depend on a choice of string structure. We illustrate nontrivial dependence on the string structure for a 16-dimensional representation of the Klein 4-group.