Gukov-Pei-Putrov-Vafa conjecture for $SU(N)/{\mathbb{Z}}_m$

Abstract

In our earlier work, we studied the $\hat{Z}$-invariant(or homological blocks) for $SO(3)$ gauge group and we found it to be same as ${\hat{Z}}^{SU(2)}$. This motivated us to study the $\hat{Z}$-invariant for quotient groups $SU(N)/{\mathbb{Z}}_m$, where $m$ is some divisor of $N$. Interestingly, we find that $\hat{Z}$-invariant is independent of $m$.

16 pages, 4 figures. v2: added a conjecture in section 4, added a point in conclusion