Abstract

Following hal.archives-ouvertes.fr/hal-03502404v3 I will show how partial theta series, i.e. functions of the form$\Theta(\tau) := \sum_{n>0} f(n) e^{{i\pi n}2 \tau/M}$ with $f : Z \to C$ an $M$-periodic function (or the product of a power of $n$ by such function), give rise to divergent asymptotic series at every rational point of the boundary of their domain of definition $\{Im\tau>0\}$. I will 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. Écalle’s “Alien calculus” allows one to encode this phenomenon in a kind of “Bridge equation”. Interesting examples stem from the study of Gukov-Pei-Putrov-Vafa invariants and Witten-Reshetikhin-Turaev invariants for the Poincaré homology sphere (cf. [Gukov-Putrov-Marino, arXiv:1605.07615]) or more generally Seifert homology 3-spheres ([Andersen-Mistegård, J. Lond. Math. Soc. 2022]).