# Higher depth quantum modular forms and plumbed $3$-manifolds

## TL;DR

This paper explores new quantum invariants of plumbed 3-manifolds, showing that certain series are depth two quantum modular forms, advancing understanding of their structure and relation to WRT invariants.

## Contribution

It proves that the series (q) for specific 3-manifolds are depth two quantum modular forms, providing new insights into their mathematical properties.

## Key findings

- (q) is a depth two quantum modular form on  for positive definite unimodular plumbing matrices.
- The study confirms conjectural links between -series and WRT invariants of plumbed 3-manifolds.
- Results extend the understanding of quantum modularity in the context of 3-manifold invariants.

## Abstract

In this paper we study new invariants $\widehat{Z}_{\boldsymbol{a}}(q)$ attached to plumbed $3$-manifolds that were introduced by Gukov, Pei, Putrov, and Vafa. These remarkable $q$-series at radial limits conjecturally compute WRT invariants of the corresponding plumbed $3$-manifold. Here we investigate the series $\widehat{Z}_{0}(q)$ for unimodular plumbing ${\tt H}$-graphs with six vertices. We prove that for every positive definite unimodular plumbing matrix, $\widehat{Z}_{0}(q)$ is a depth two quantum modular form on $\mathbb{Q}$.