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

Abstract

In this paper we study new invariants ${\hat{Z}}_{\mathbf{a}}(q)$ 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 ${\hat{Z}}_0(q)$ for unimodular plumbing $\mathtt{H}$-graphs with six vertices. We prove that for every positive definite unimodular plumbing matrix, ${\hat{Z}}_0(q)$ is a depth two quantum modular form on $\mathbb{Q}$.

22 pages; 1 figure

References in corpus (3)

• 3d Modularity
• Quantum Knot Invariant for Torus Link and Modular Forms
• On the Quantum Invariant for the Brieskorn Homology Spheres

Cited by in corpus (9)

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• Higher rank $\hat{Z}$ and $F_K$
• Branches, quivers, and ideals for knot complements
• Cobordism invariants from BPS q-series
• Learning BPS Spectra and the Gap Conjecture
• Quantum Modular $\hat{Z}{}^G$-Invariants
• Witten-Reshetikhin-Turaev Invariants, Homological Blocks, and Quantum Modular Forms for Unimodular Plumbing H-Graphs