Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 034, 20 pages      arXiv:2110.10958      https://doi.org/10.3842/SIGMA.2022.034

Witten-Reshetikhin-Turaev Invariants, Homological Blocks, and Quantum Modular Forms for Unimodular Plumbing H-Graphs

Akihito Mori and Yuya Murakami
Mathematical Institute, Tohoku University, 6-3, Aoba, Aramaki, Aoba-Ku,Sendai 980-8578, Japan

Received November 23, 2021, in final form April 28, 2022; Published online May 07, 2022

Abstract
Gukov-Pei-Putrov-Vafa constructed $ q $-series invariants called homological blocks in a physical way in order to categorify Witten-Reshetikhin-Turaev (WRT) invariants and conjectured that radial limits of homological blocks are WRT invariants. In this paper, we prove their conjecture for unimodular H-graphs. As a consequence, it turns out that the WRT invariants of H-graphs yield quantum modular forms of depth two and of weight one with the quantum set $ \mathbb{Q} $. In the course of the proof of our main theorem, we first write the invariants as finite sums of rational functions. We second carry out a systematic study of weighted Gauss sums in order to give new vanishing results for them. Combining these results, we finally prove that the above conjecture holds for H-graphs.

Key words: quantum invariants; Witten-Reshetikhin-Turaev invariants; homological blocks; quantum modular forms; plumbed manifolds; false theta funcitons; Gauss sums.

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