sigma model with target $S^3$ and B-field of strength $24$). So the puzzle requires constructing a torsion-valued deformation-invariant of SQFTs that is more sensitive than the elliptic genus.

The motivation for the puzzle comes from the theory of Topological Modular Forms (TMF) described for example in [@MR1989190; @MR3223024]. Based on suggestions from [@MR992209], it is conjectured in [@MR2079378; @MR2742432] that every SQFT defines a class in TMF, invariant under deformations of the SQFT. Indeed, [@MR2079378; @MR2742432] conjecture that this TMF class exactly captures the deformation class of the corresponding SQFT. The TMF-valued invariant of an SQFT refines the usual modular-form valued elliptic index. It is known that the TMF-valued invariant of $S^3$ (with minimal nonzero B-field) has exact order $24$, hence the puzzle.

In this paper we will propose a solution to this puzzle. Let ${\mathcal B}$ be an SCFT which, like the one in Puzzle \[mainpuzzle\], has gravitational anomaly $c_R - c_L = 3/2$.[^1] If ${\mathcal B}$ is not initially conformal, flow it to the IR before proceeding. Build:

1.  a “generalized mock modular form $f_1$” with source equal to the torus one-point function of the supersymmetry generator of ${\mathcal B}$.[^2] The $q$-expansion of $f_1$ is not determined by 