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 {-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.