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We construct deformation invariants of $2|1$-dimensional Euclidean fieldtheories valued in a cohomology theory approximating topological modular forms.This implies several results anticipated by Stolz and Teichner and gives thefirst torsion invariants of field theories valued in $\pi_*{\rm TMF}$. Theframework leads to a version of the elliptic Euler class described in terms offield-theoretic data, as well as a partial construction of the supersymmetricsigma model with target a string manifold.

How do field theories detect the torsion in topological modular forms?