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We provide a differential cocycle model for elliptic cohomology with complexcoefficients and use analytic methods to construct a cocycle representative forthe Witten class in this language. Our motivation stems from the conjecturalconnection between 2-dimensional field theories and elliptic cohomologyoriginally due to G. Segal and E. Witten. The specifics of our constructionsare informed by the work of S. Stolz and P. Teichner on super Euclidean fieldtheories and K. Costello's construction of the Witten genus using perturbativequantization. As a warm-up, we prove analogous results for supersymmetricquantum mechanics and K-theory with complex coefficients.

Perturbative sigma models, elliptic cohomology and the Witten genus