A framework for proving quantum modularity: Application to Witten's asymptotic expansion conjecture

TL;DR

This paper develops new techniques to analyze quantum invariants and false theta functions, extending previous results to more complex 3-manifolds and broadening the scope of quantum modularity proofs.

Contribution

It introduces a Poisson summation formula with signature and a modular series framework to handle multivariable integrals, advancing quantum topology and number theory.

Findings

Extended asymptotic analysis to negative definite plumbed 3-manifolds

Established quantum modularity for a wider class of false theta functions

Unified approach to quantum modularity for various special functions

Abstract

We address two linked problems at the interface of quantum topology and number theory: deriving asymptotic expansions of the Witten--Reshetikhin--Turaev invariants for 3-manifolds and establishing quantum modularity of false theta functions. Previous progress covers Seifert homology 3-spheres for the former and rank-one cases for the latter, both of which rely on single-variable integral representations. We extend these results to negative definite plumbed 3-manifolds and to general false theta functions, respectively. We address this limitation by developing two techniques: a Poisson summation formula with signature and a framework of modular series, both of which enable a precise and explicit analysis of multivariable integral representations. As further applications, our method yields a unified approach to proving quantum modularity for false theta functions, indefinite theta functions, and for Eisenstein series of odd weight.