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

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.