  We construct a map from $d|1$-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 parameterized
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 $\mathcal{N}=(0,1)$ supersymmetric
quantum field theories are weak modular forms, following a suggestion of Stolz
and Teichner.
