Supergroups, q-Series and 3-Manifolds

We introduce supergroup analogs of 3-manifold invariants \({\widehat{Z}}\), also known as homological blocks, which were previously considered for ordinary compact semisimple Lie groups. We focus on superunitary groups and work out the case of SU(2|1) in details. Physically these invariants are realized as the index of BPS states of a system of intersecting fivebranes wrapping a 3-manifold in M-theory. As in the original case, the homological blocks are q-series with integer coefficients. We provide an explicit algorithm to calculate these q-series for a class of plumbed 3-manifolds and study quantum modularity and resurgence properties for some particular 3-manifolds. Finally, we conjecture a formula relating the \({\widehat{Z}}\) invariants and the quantum invariants constructed from a non-semisimple category of representation of the unrolled version of a quantum supergroup.