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

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