Non-semisimple topological field theory and Z-invariants from osp(1 2)

We construct three dimensional non-semisimple topological field theories from the unrolled quantum group of the Lie superalgebra osp(1 2). More precisely, the quantum group depends on a root of unity q=e^2 π √-1r, where r is a positive integer greater than $2$, and the construction applies when r is not congruent to $4$ modulo $8$. The algebraic result which underlies the construction is the existence of a relative modular structure on the non-finite, non-semisimple category of weight modules for the quantum group. We prove a Verlinde formula which allows for the computation of dimensions and Euler characteristics of topological field theory state spaces of unmarked surfaces. When r is congruent to ± 1 or ± 2 modulo $8$, we relate the resulting $3$-manifold invariants with physicists' Z-invariants associated to osp(1 2). Finally, we establish a relation between Z-invariants associated to sl(2) and osp(1 2) which was conjectured in the physics literature.