Non-archimedean cylinder counts are logarithmic Gromov-Witten invariants
Abstract
We establish a comparison result relating non-archimedean cylinder counts and logarithmic cylinder counts in a smooth affine log Calabi-Yau variety. Using the decomposition theorem and the gluing formula from log Gromov-Witten theory, we can express logarithmic cylinder counts in terms of wall type invariants. As a corollary, we show that in the surface case the non-archimedean scattering diagram from Keel-Yu and the logarithmic scattering diagram from Gross-Siebert coincide, and deduce that the two mirror constructions agree. Along the way, we prove the exponential formula, expressing the non-archimedean wall-crossing function as the exponential of a generating series of punctured log Gromov-Witten invariants. This provides the first explicit formula relating counts of non-archimedean curves with boundary to punctured log invariants.
Keywords
Cite
@article{arxiv.2510.18319,
title = {Non-archimedean cylinder counts are logarithmic Gromov-Witten invariants},
author = {Thorgal Hinault and Tony Yue YU},
journal= {arXiv preprint arXiv:2510.18319},
year = {2025}
}