Supersymmetric field theories and the elliptic index theorem with complex coefficients

TL;DR

This paper develops a cocycle model for elliptic cohomology with complex coefficients using 2D quantum field theory, providing a rigorous construction of cocycles and connecting physics with elliptic index theorems.

Contribution

It introduces a new cocycle model for elliptic cohomology with complex coefficients derived from quantum field theory methods, linking physics and topology.

Findings

Constructs the complexified string orientation of elliptic cohomology.

Establishes the equivalence of two pushforwards from different quantum field theories.

Provides a physical interpretation of the elliptic index theorem with complex coefficients.

Abstract

We present a cocycle model for elliptic cohomology with complex coefficients in which methods from 2-dimensional quantum field theory can be used to rigorously construct cocycles. For example, quantizing a theory of vector bundle-valued fermions yields a cocycle representative of the elliptic Thom class. This constructs the complexified string orientation of elliptic cohomology, which determines a pushfoward for families of rational string manifolds. A second pushforward is constructed from quantizing a supersymmetric $\sigma$-model. These two pushforwards agree, giving a precise physical interpretation for the elliptic index theorem with complex coefficients. This both refines and supplies further evidence for the long-conjectured relationship between elliptic cohomology and 2-dimensional quantum field theory. Analogous methods in supersymmetric mechanics recover path integral constructions of the Mathai--Quillen Thom form in complexified ${\rm KO}$-theory and a cocycle representative of the $\hat{A}$-class for a family of oriented manifolds.