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

TL;DR

This paper proves a conjecture linking homological blocks and Witten-Reshetikhin-Turaev invariants for unimodular H-graphs, revealing their connection to quantum modular forms and advancing understanding in quantum topology.

Contribution

It establishes the conjecture for unimodular H-graphs, showing that homological blocks' radial limits are WRT invariants and connecting them to quantum modular forms.

Findings

WRT invariants of H-graphs are quantum modular forms of depth two and weight one.

Homological blocks' radial limits equal WRT invariants for unimodular H-graphs.

New vanishing results for weighted Gauss sums were derived.

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.