This article is cited in 2 papers

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

Akihito Mori, Yuya Murakami

Mathematical Institute, Tohoku University, 6-3, Aoba, Aramaki, Aoba-Ku, Sendai 980-8578, Japan

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.

Keywords: quantum invariants, Witten–Reshetikhin–Turaev invariants, homological blocks, quantum modular forms, plumbed manifolds, false theta funcitons, Gauss sums.

MSC: 57K31, 57K10, 57K16, 11F27, 11L05, 11T24

Received: November 23, 2021; in final form April 28, 2022; Published online May 7, 2022

Language: English

DOI: 10.3842/SIGMA.2022.034

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ArXiv: 2110.10958