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

TL;DR

This paper investigates the $ ext{hat}Z$-invariant for quotient groups $SU(N)/Z_m$, revealing its independence from $m$, and extends previous work on $SO(3)$ and $SU(2)$ gauge groups.

Contribution

It demonstrates that the $ ext{hat}Z$-invariant remains unchanged for different divisors $m$ of $N$ in quotient groups $SU(N)/Z_m$, generalizing earlier findings.

Findings

$ ext{hat}Z$-invariant is independent of $m$ for $SU(N)/Z_m$

Extension of previous results from $SO(3)$ and $SU(2)$

Supports the conjecture relating $ ext{hat}Z$-invariants across quotient groups

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 $m$.