and supersymmetry encoding the structure constants of $\mathfrak{g}_2$ is not nullhomotopic.

3.  Show that the product of $\overline{\operatorname{Fer}}(\mathfrak{g}_2)$ with a (standard) $S^1$ sigma model is not nullhomotopic.

A torsion invariant of SQFTs {#sec.general}
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Gravitational anomalies and spectator fermions {#sec.anomaly}
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For a quantum field theory to be [*gravitationally nonanomalous*]{}, its partition function must be valued in numbers (as opposed to a section of some line bundle on the moduli space of spacetimes); it must have a well-defined Hilbert space (as opposed to a section of some gerbe); and so on for higher-codimensional data [@FreedTeleman2012]. In the $(1+1)$-dimensional SQFT case, the [*elliptic genus*]{} $Z_{RR}$ is by definition the partition function on flat tori with nonbounding, aka Ramond–Ramond aka periodic–periodic, spin structures. The moduli space of Ramond–Ramond flat tori is three-real-dimensional — in addition to the complex and anticomplex parameters $(\tau,\bar\tau)$, there is also a “size” parameter — but we will generally compute in the IR aka large-torus limit. In this limit, the partition function of a nonanomalous $(1+1)$-dimensional SQFT will definitely be modular for the full $\mathrm{SL}(2,{{\mathds}Z})$ with weight $(0,0)$ and no multiplier.[^7] Indeed, modularity is transparent from the path-integral description.[^8]

The SQFT in Puzzle \[mainpuzzle\], and more generally any 