Supersymmetric field theories and 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 pushforward 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 $\text{KO}$–theory and a cocycle representative of the $Â$–class for a family of oriented manifolds.