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Non-archimedean cylinder counts are logarithmic Gromov-Witten invariants

Thorgal Hinault, Tony Yue YU

TL;DR

This work establishes a precise bridge between non-archimedean and logarithmic enumerative geometries for smooth affine log Calabi–Yau varieties. It proves an explicit decomposition of non-archimedean cylinder counts $N(S,A)$ into logarithmic cylinder counts $N_{\boldsymbol{\tau}}$ via wall-type invariants, encapsulated by $N(S,A)=\sum_{\boldsymbol{\tau}} k_{\tau} N_{\boldsymbol{\tau}}$, and introduces an exponential wall-crossing formula $f_x^{\mathrm{an}}(t,z)=f_x^{\log}(t,z)$ for all $x$ in the essential skeleton, thereby equating the non-archimedean and logarithmic scattering diagrams in the surface case. The paper leverages the decomposition/gluing machinery of log Gromov–Witten theory to connect boundary-restricted non-archimedean curves with punctured log invariants, concluding that the two mirror constructions (Keel–Yu and Gross–Siebert) agree for affine log Calabi–Yau surfaces. This yields a principled method to translate counts across frameworks and provides a first explicit formula linking non-archimedean curve counts with boundary to punctured log invariants. The results pave the way for broader comparisons and higher-dimensional generalizations in subsequent work.

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.

Non-archimedean cylinder counts are logarithmic Gromov-Witten invariants

TL;DR

This work establishes a precise bridge between non-archimedean and logarithmic enumerative geometries for smooth affine log Calabi–Yau varieties. It proves an explicit decomposition of non-archimedean cylinder counts into logarithmic cylinder counts via wall-type invariants, encapsulated by , and introduces an exponential wall-crossing formula for all in the essential skeleton, thereby equating the non-archimedean and logarithmic scattering diagrams in the surface case. The paper leverages the decomposition/gluing machinery of log Gromov–Witten theory to connect boundary-restricted non-archimedean curves with punctured log invariants, concluding that the two mirror constructions (Keel–Yu and Gross–Siebert) agree for affine log Calabi–Yau surfaces. This yields a principled method to translate counts across frameworks and provides a first explicit formula linking non-archimedean curve counts with boundary to punctured log invariants. The results pave the way for broader comparisons and higher-dimensional generalizations in subsequent work.

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.
Paper Structure (29 sections, 38 theorems, 78 equations, 4 figures)

This paper contains 29 sections, 38 theorems, 78 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Motivations
  3. Main results
  4. Non-archimedean cylinder counts are logarithmic Gromov-Witten invariants
  5. Comparison of mirror symmetry constructions for affine log Calabi-Yau surfaces
  6. Future directions
  7. Non-archimedean and logarithmic cylinders
  8. Affine log Calabi-Yau pair
  9. Assumptions
  10. Skeleton and essential skeleton
  11. Logarithmic stable maps in $X_B/B$
  12. Walls and balancing condition
  13. Non-archimedean walls and transverse spines
  14. Logarithmic walls and broken lines
  15. Cylinder counts
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $U$ be a connected smooth affine log Calabi-Yau variety satisfying the assumptions of subsec:log-setup. Let $S$ be a cylinder spine in $\mathop{\mathrm{Sk}}\nolimits (U)$ of type $\beta = (A,\boldsymbol{\mathrm{u}} )$. Then the non-archimedean cylinder count $N(S,A)$ decomposes as a sum of logar where the sum is over decorated tropical cylinder types $\boldsymbol{\tau}$ of total curve class $A

Figures (4)

  • Figure 1: Typical curves appearing on the generic and special fiber of the point constrained moduli spaces constructed in § \ref{['subsec:family-moduli-space']}. The point constraint is at the marked point $p_i$, while $p_1$ and $p_2$ are mapped to specified components of the boundary $D$.
  • Figure 2: A typical cylinder spine (in solid red). A tropical cylinder type associated to this spine has wall types attached to the bending vertices (in dashed red).
  • Figure 3: The refined geometric moduli space $\mathcal{M} (S)_T$ is obtained by selecting vertical components in the special fiber, and considering the horizontal components which intersect them.
  • Figure 4: A typical infinitesimal cylinder spine (in solid red). To make it a tropical curve coming from a stable map, wall types are attached to the bending vertex (in dashed red).

Theorems & Definitions (81)

  • Theorem 1.1: Main comparison, \ref{['thm:main-comparison']}
  • Proposition 1.2: \ref{['prop:infinitesimal-cylinder-counts-to-log-counts']}
  • Theorem 1.3: Exponential formula, \ref{['thm:exponential-formula']}
  • Corollary 1.4: \ref{['coro:mirror-algebras']}
  • Definition 2.1: Log Calabi-Yau variety
  • Remark 2.6
  • Definition 2.11: Spine of a non-archimedean stable map
  • Lemma 2.13
  • proof
  • Definition 2.15: Simple type, spine type
  • ...and 71 more