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

TL;DR

This paper extends the explicit construction of $$-series invariants, known as homological blocks, to the $SO(3)$ and $OSp(1|2)$ groups, revealing their relation to $SU(2)$ invariants through variable changes.

Contribution

It provides explicit $q$-series formulas for $$-invariants of $SO(3)$ and $OSp(1|2)$, expanding the understanding of homological blocks for these groups.

Findings

Explicit $q$-series forms for $$-invariants of $SO(3)$ and $OSp(1|2)$

Relation between $SU(2)$ and these invariants via variable change

Enhanced understanding of homological blocks for supergroups and orthogonal 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 ortho-symplectic 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 $SO(3)$ & $OSp(1|2)$ link invariants plays a crucial role in explicitly writing the $q$-series.