$\hat{Z}$- 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 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.

arXiv comment18 pages, 2 figures; minor changes; matches the published version

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Cited by in corpus (1)

• Quantum Modular $\hat{Z}{}^G$-Invariants