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

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 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.

Non-Semisimple TQFT's and BPS q-Series

“…The conjecture above is an analogue of the of the conjecture relating WRT invariant with the limits of Z s that was formulated in [43,44]. It was proven for certain families of 3-manifolds in [3,36,41,60]. Some elements of the latter conjecture can trace its origin to the work of Lawrence and Zagier [58].…”