Higher depth quantum modular forms and plumbed 3-manifolds

Bringmann, Kathrin; Mahlburg, Karl; Milas, Antun

doi:10.1007/s11005-020-01310-z

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Higher depth quantum modular forms and plumbed 3-manifolds

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Milas, Antun
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1007/s11005-020-01310-z
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arXiv:1906.10722.pdf
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Bringmann, K., Mahlburg, K., & Milas, A. (2020). Higher depth quantum modular forms and plumbed 3-manifolds. Letters in Mathematical Physics, 110(10), 2675-2702. doi:10.1007/s11005-020-01310-z.

Cite as: https://hdl.handle.net/21.11116/0000-0006-BD9F-3

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
$\mathbb{Q}$.