$${\hat{Z}}$$-Invariant for SO(3) and OSp(1|2) Groups

Abstract

Three-manifold invariants $${\hat{Z}}$$ (“Z-hat”), also known as homological blocks, are q-series with integer coefficients. Explicit q-series form for $${\hat{Z}}$$ is known for SU(2) group, supergroup SU(2|1) and orthosymplectic supergroup OSp(2|2). We focus on $${\hat{Z}}$$ for SO(3) group and orthosymplectic supergroup OSp(1|2) in this paper. Particularly, the change of variable relating SU(2) link invariants to the SO(3) and OSp(1|2) link invariants plays a crucial role in explicitly writing the q-series.