Invariants of r-spin TQFTs and non-semisimplicity

Abstract

For a positive integer r, an r-spin topological quantum field theory is a 2-dimensional TQFT with tangential structure given by the r-fold cover of ${\mathrm{SO}}_2$. In particular, such a TQFT assigns a scalar invariant to every closed r-spin surface Σ. Given a sequence of scalars indexed by the set of diffeomorphism classes of all such Σ, we construct a symmetric monoidal category $C$ and a $C$-valued r-spin TQFT which reproduces the given sequence. We also determine when such a sequence arises from a TQFT valued in an abelian category with finite-dimensional Hom spaces. In particular, we construct TQFTs with values in super vector spaces that can distinguish all diffeomorphism classes of r-spin surfaces, and we show that the Frobenius algebras associated to such TQFTs are necessarily non-semisimple.