Asymptotic behavior of partial and false theta functions arising from Jacobi forms and regularized characters

TL;DR

This paper investigates the asymptotic behavior of partial and false theta functions from Jacobi forms, revealing Stokes' phenomenon and applying findings to regularized characters and quantum dimensions in conformal field theory.

Contribution

It provides new asymptotic results for theta functions and extends understanding of regularized quantum dimensions in vertex operator algebras.

Findings

Revealed Stokes' phenomenon in theta functions

Derived asymptotic expansions for regularized characters

Extended known results on quantum dimensions

Abstract

We prove several asymptotic results for partial and false theta functions arising from Jacobi forms, as the modular variable $\tau$ tends to $0$ along the imaginary axis, and the elliptic variable $z$ is unrestricted in the complex plane. We observe that these functions exhibit Stokes' phenomenon - the asymptotic behavior of these functions sharply differs depending on where the elliptic variable $z$ is located within the complex plane. We apply our results to study the asymptotic expansions of regularized characters and quantum dimensions of the $(1,p)$-singlet vertex operator algebra coming from conformal field theory. This, in particular, recovers and extends several known results pertaining to regularized quantum dimensions, which served as a main source of motivation.