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

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