We construct three-dimensional non-semisimple topological field theories from the unrolled quantum group of the Lie superalgebra \(\mathfrak {osp}(1 \vert 2)\) . More precisely, the quantum group depends on a root of unity \(q=e^{\frac{2 \pi \sqrt{-1}}{r}}\) , 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 \(\pm 1\) or \(\pm 2\) modulo 8, we relate the resulting 3-manifold invariants with physicists’ \(\widehat{Z}\) -invariants associated to \(\mathfrak {osp}(1 \vert 2)\) . Finally, we establish a relation between \(\widehat{Z}\) -invariants associated to \(\mathfrak {sl}(2)\) and \(\mathfrak {osp}(1 \vert 2)\) which was conjectured in the physics literature.
