Quantum modular forms and plumbing graphs of 3-manifolds

2018/10/12 by Bringmann, Kathrin, Mahlburg, Karl, Milas, Antun
#11F27 #11F37 #14N35 #57M27 #57R56 #FOS: Mathematics #Number Theory (math.NT) #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.1810.05612

Abstract

In this paper, we study quantum modular forms in connection to quantum invariants of plumbed 3-manifolds introduced recently by Gukov, Pei, Putrov, and Vafa. We explicitly compute these invariants for any 3-leg star plumbing graphs whose associated matrix is unimodular and positive definite. For these graphs we confirm a quantum modularity conjecture of Gukov. We also analyze the invariants for general n-leg star graphs with unimodular plumbing matrices, and prove that they can be expressed as linear combinations of quantum modular forms.

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