Resurgence and Partial Theta Series

TL;DR

This paper studies partial theta series with periodic coefficients, exploring their asymptotic, summability, and resurgence properties, and linking their modularity to Stokes phenomena and discrete Fourier transforms.

Contribution

It introduces explicit formulas for Borel transforms of these series and reveals a novel connection between quantum modularity and Fourier analysis.

Findings

Explicit Borel transform formulas for partial theta series

Connection between quantum modularity and Stokes phenomena

Role of Discrete Fourier Transform in modularity analysis

Abstract

We consider partial theta series associated with periodic sequences of coefficients, of the form $\Theta(\tau) := \sum_{n>0} n^\nu f(n) e^{i\pi n^2\tau/M}$, with $\nu$ non-negative integer and an $M$-periodic function $f : \mathbb{Z} \rightarrow \mathbb{C}$. Such a function is analytic in the half-plane $\{Im(\tau)>0\}$ and as $\tau$ tends non-tangentially to any $\alpha\in\mathbb{Q}$, a formal power series appears in the asymptotic behaviour of $\Theta(\tau)$, depending on the parity of $\nu$ and $f$. We discuss the summability and resurgence properties of these series by means of explicit formulas for their formal Borel transforms, and the consequences for the modularity properties of $\Theta$, or its ``quantum modularity'' properties in the sense of Zagier's recent theory. The Discrete Fourier Transform of $f$ plays an unexpected role and leads to a number-theoretic analogue of \'Ecalle's ``Bridge Equations''. The motto is: (quantum) modularity = Stokes phenomenon + Discrete Fourier Transform.