Resurgent Lambert series from Feynman and beyond

David Broadhurst; Daniele Dorigoni

mathematical physics

Resurgent Lambert series from Feynman and beyond

arXiv:2607.14020

summary

The paper analyzes Lambert series arising from multi‑loop Feynman diagrams, uses resurgence methods to control their singular limit as |q|→1, and extends the approach to modular resurgent structures in topological‑string observables.

Abstract

Lambert series of the form $\sum_{n > 0} a (n) q^{n} / (1 - q^{n})$ are ubiquitous in mathematical physics. In particular, 2-loop sunrise and 3-loop banana Feynman diagrams yield Lambert series with $a (n)$ of the form $I ¨ (n) / n^{s}$ where $I ¨ (n)$ is a Dirichlet character. Resurgence concerns the singular limit as $∣ q ∣$ approaches 1. In the Feynman cases we can control this limit, obtaining rapidly convergent expressions, since the Lambert series are iterated integrals of holomorphic Eisenstein series twisted by a character. We generalize this result, to include modular resurgent structures found in topological-string observables.

8 pages, talk given at Loops and Legs 2026. Version accepted in Proceedings of Science (LL2026) 020

Topics & keywords

#lambert series #resurgence #feynman diagrams #eisenstein series #modular forms #topological stringsLambert seriesDirichlet characterEisenstein seriesresurgencesunrise diagrambanana diagrammodular resurgent structures