Higher depth quantum modular forms and plumbed 3-manifolds
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Higher depth quantum modular forms and plumbed 3-manifolds Kathrin Bringmann1 · Karl Mahlburg2 · Antun Milas3,4 Received: 4 July 2019 / Revised: 9 March 2020 / Accepted: 20 June 2020 © Springer Nature B.V. 2020
Abstract 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 0 (q) for unimodular plumbing H-graphs with six vertices. We prove that for every positive definite unimodular plumbing matrix, Z 0 (q) is a depth two quantum modular form on Q. Keywords Quantum invariants · Plumbing graphs · Quantum modular forms Mathematics Subject Classification 11F27 · 11F37 · 14N35 · 57M27 · 57R56
The research of the first author was supported by the Alfried Krupp Prize for Young University Teachers of the Krupp foundation, and the research leading to these results receives funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement no. 335220 - AQSER. The third author was supported by NSF-DMS Grant 1601070 and a stipend from the Max Planck Institute for Mathematics, Bonn.
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Antun Milas [email protected] Kathrin Bringmann [email protected] Karl Mahlburg [email protected]
1
Department of Mathematics and Computer Science, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany
2
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA
3
Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
4
Present Address: Department of Mathematics and Statistics, SUNY-Albany, Albany, NY 12222, USA
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K. Bringmann et al.
1 Introduction and statement of results A quantum modular form is a complex-valued function defined on Q or a subset thereof, called the quantum set, that exhibits modular-like transformation properties up to an obstruction term with “nice” analytic properties (for instance, it can be extended to a real analytic function on some open subset of R). Quantum modular forms were introduced by Zagier in [23], where he described several non-trivial examples. They have appeared in several areas including quantum invariants of knots and 3-manifolds [16–19], mock modular forms [13], meromorphic Jacobi forms [8], mathematical physics [12], partial and false theta functions [7], and representation theory [7,11]. Motivated on the one hand by the concept of higher depth mock modular forms and on the other hand by the appearance of higher rank false theta functions in representation theory, Kaszian and two of the authors [4] defined so-called higher depth quantum modular forms and gave an infinite family of examples coming from characters of representations of vertex algebras. If the depth is two, these functions satisfy f (τ ) − (cτ + d)−k f (γ τ ) ∈ Q1 O(R),
γ =
a c
b d
∈ SL2 (Z),
where Q1 is the space of quantum modular forms and O(R) is the
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