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