Supersymmetric field theories and the elliptic index theorem with complex coefficients

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 of vector bundle-valued fermions yields representative the Thom class. This constructs complexified string orientation cohomology, determines pushfoward families rational manifolds. A second pushforward is constructed supersymmetric $\sigma$-model. These two pushforwards agree, giving precise physical interpretation index theorem coefficients. both refines and supplies further evidence long-conjectured relationship between theory. Analogous mechanics recover path integral constructions Mathai--Quillen form ${\rm KO}$-theory $\hat{A}$-class family oriented