Higher depth quantum modular forms and plumbed 3-manifolds

In this paper, we study new invariants Z^ _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 Z^ (q) for unimodular plumbing H-graphs with six vertices. We prove that for every positive definite unimodular plumbing matrix, Z^ (q) is a depth two quantum modular form on Q.