which we characterize in a precise manner in terms of the torus one-point function of the supersymmetry generator in a compact “boundary SQFT”. We use this construction to argue that the torus one-point function of the supersymmetry generator in a nullhomotopic SQFT is the source of a holomorphic anomaly equation for a generalized mock modular form. We then argue that the coefficients of this generalized mock modular form are (even) integral, up to a correction arising as a type of “mod 2 index” of the boundary SQFT. It follows that, if solutions of the holomorphic anomaly equation fail to have integral (plus correction) mock modular parts, the SQFT cannot be nullhomotopic. This is the justification for our invariants. We then recast our invariant in homotopy-theoretic terms, where it becomes the “secondary invariant” of [@MR3278648].

To illustrate our proposed invariant, we focus on two families of examples. First, in Section \[sec.cigar\], we study the sigma models with target $S^1$ and the “cigar.” We start by reviewing the $S^1$ sigma models, with an emphasis on the role that the target-space spin structure plays on the behaviour of the model. This provides a chance to illustrate the mod-2 index, and allows us to 