compute our invariant for the $T^3$ sigma model with its Lie group framing. We then analyze the “cigar,” which is a noncompact manifold with “boundary” $S^1$, and demonstrate explicitly that the corresponding sigma model enjoys our predicted holomorphic anomaly and integrality.

The second set of examples, which we study in Section \[sec.S3\], are the ones from [@WittenTMF]: the ${\mathcal N}=(0,1)$ sigma model with target the round $S^3$ and with B-field of strength $k$. We first warm up with the $k=1$ case of Puzzle \[mainpuzzle\], and then study the general case. In all cases, we find that our invariant is precisely $k \pmod {24}$, showing that the $S^3_k$ sigma model is nullhomotopic if and only if $k = 0 \pmod {24}$. We then mention a few related constructions and puzzles: we build an antiholomorphic SCFT of central charge $c_R = 27/2$ which we expect to represent the 3-torsion element in $\pi_{27}\mathrm{TMF}$; and we speculate that $S^3_k$ is “flavoured-nullhomoptic” for all $k$, with the nullhomotopy given by a certain “trumpet” geometry with ${\mathcal N}=(0,4)$ supersymmetry.

Let us end this introduction by emphasizing that we expect our invariant captures only some of the torsion in the space of ${\mathcal N}=(0,1)$ SQFTs.[^5] It is 