Witten–Reshetikhin–Turaev invariants and homological blocks for plumbed homology spheres

In this paper, we prove a conjecture by Gukov–Pei–Putrov–Vafa for a wide class of plumbed $3$-manifolds. Their conjecture states that Witten–Reshetikhin–Turaev (WRT) invariants are radial limits of homological blocks, which are $q$-series introduced by them for plumbed $3$-manifolds with negative definite linking matrices. The most difficult point in our proof is to prove the vanishing of weighted Gauss sums that appear in coefficients of negative degree in asymptotic expansions of homological blocks. To deal with it, we develop a new technique for asymptotic expansions, which enables us to compare asymptotic expansions of rational functions and false theta functions related to WRT invariants and homological blocks, respectively. In our technique, our vanishing results follow from holomorphy of such rational functions.

Keywords

quantum invariants, quantum modular forms, false theta series, Gauss sums, asymptotic expansions

2010 Mathematics Subject Classification

11F27, 11L05, 11T24

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The author is supported by JSPS KAKENHI grants JP20J20308 and JP23KJ1675.

Received 25 May 2022

Accepted 3 April 2024

Published 15 July 2024