Tropical correspondence for smooth del Pezzo log Calabi-Yau pairs

TL;DR

This paper establishes a correspondence between genus zero logarithmic Gromov-Witten invariants of certain del Pezzo pairs and the wall structures in the Gross-Siebert program, linking enumerative geometry with mirror symmetry constructions.

Contribution

It introduces a novel connection between logarithmic Gromov-Witten invariants and the wall structures in the Gross-Siebert mirror symmetry framework for smooth del Pezzo pairs.

Findings

Logarithm of wall functions encodes Gromov-Witten invariants.

Wall structures correspond to enumerative invariants in the Calabi-Yau setting.

Provides a new computational approach for invariants using mirror symmetry.

Abstract

Consider a log Calabi-Yau pair $(X,D)$ consisting of a smooth del Pezzo surface $X$ of degree $\geq 3$ and a smooth anticanonical divisor $D$. We prove a correspondence between genus zero logarithmic Gromov-Witten invariants of $X$ intersecting $D$ in a single point with maximal tangency and the consistent wall structure appearing in the dual intersection complex of $(X,D)$ from the Gross-Siebert reconstruction algorithm. More precisely, the logarithm of the product of functions attached to unbounded walls in the consistent wall structure gives a generating function for these invariants.