Non-archimedean cylinder counts are logarithmic Gromov-Witten invariants

TL;DR

This paper establishes a deep connection between non-archimedean and logarithmic cylinder counts in log Calabi-Yau varieties, showing their equivalence in certain cases and linking mirror symmetry constructions.

Contribution

It provides the first explicit formula relating non-archimedean curve counts with boundary to punctured log Gromov-Witten invariants and proves the exponential formula for wall-crossing functions.

Findings

Non-archimedean and logarithmic scattering diagrams coincide in surface cases.

Logarithmic cylinder counts can be expressed via wall invariants.

The exponential formula relates wall-crossing functions to punctured log 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.