Chern characters for supersymmetric field theories

We construct a map from dj1–dimensional Euclidean field theories to complexified K–theory when d = 1 and complex-analytic elliptic cohomology when d = 2. This provides further evidence for the Stolz–Teichner program, while also identifying candidate geometric models for Chern characters within their framework. The construction arises as a higher-dimensional and parametrized generalization of Fei Han’s realization of the Chern character in K–theory as dimensional reduction for 1|1–dimensional Euclidean field theories. In the elliptic case, the main new feature is a subtle interplay between the geometry of the super moduli space of 2|1–dimensional tori and the derived geometry of complex-analytic elliptic cohomology. As a corollary, we obtain an entirely geometric proof that partition functions of N = (0,1) supersymmetric quantum field theories are weak modular forms, following a suggestion of Stolz and Teichner.