A two-dimensional, minimally Supersymmetric Quantum Field Theory is "nullhomotopic" if it can be deformed to one with spontaneous supersymmetry breaking, including along deformations that are allowed to "flow up" along RG flow lines. SQFTs modulo nullhomotopic SQFTs form a graded abelian group $SQFT_\bullet$. There are many SQFTs with nonzero index; these are definitely not nullhomotopic, and indeed represent nontorision classes in $SQFT_\bullet$. But relations to topological modular forms suggests that $SQFT_\bullet$ also has rich torsion. Based on an analysis of mock modularity and holomorphic anomalies, I will describe explicitly a "secondary invariant" of SQFTs and use it to show that a certain element of $SQFT_3$ has exact order $24$. This work is joint with D. Gaiotto and E. Witten.
- Topology Seminar