The mirror of the cubic surface
Mark Gross, Paul Hacking, Sean Keel, Bernd Siebert
TL;DR
This paper fully describes the mirror of the cubic surface using scattering diagrams and Gromov-Witten invariants, advancing the understanding of mirror symmetry for log Calabi-Yau surfaces.
Contribution
It provides a complete description of the scattering diagram and derives the mirror cubic family equation through two different methods, enriching mirror symmetry techniques.
Findings
Complete scattering diagram for the cubic surface
Explicit equation of the mirror cubic family
Application of Gromov-Witten invariants in mirror construction
Abstract
This paper expands on a remark in the paper "Mirror Symmetry for Log Calabi-Yau Surfaces I" of the first three authors of this paper, explaining fully how various constructions of the authors apply to give the mirror to the cubic surface. We give a full description of the scattering diagram associated to the cubic surface: this is a particularly nice diagram in which rays of every rational slope occur, but they may all be described. The equation of the mirror cubic family is then derived in two ways, first by using broken lines and then by using more recent constructions involving a direct calculation of Gromov-Witten invariants.
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