Intrinsic Mirror Symmetry

TL;DR

This paper constructs a ring associated with log Calabi-Yau pairs and degenerations, proposing a mirror symmetry framework where the ring's spectrum or Proj corresponds to the mirror, and proves the ring's algebraic properties.

Contribution

It introduces a new ring R defined via tropicalization and Gromov-Witten invariants, establishing its algebraic structure and proposing its role in mirror symmetry for Calabi-Yau geometries.

Findings

Ring R is associative and commutative with a unit.

Spec R and Proj R are proposed as mirrors in different degeneration cases.

Main result: R's algebraic properties are rigorously proven.

Abstract

We associate a ring R to a log Calabi-Yau pair (X,D) or a degeneration of Calabi-Yau manifolds X->B. The vector space underlying R is determined by the tropicalization of (X,D) or X->B, while the product rule is defined using punctured Gromov-Witten invariants, defined in joint work with Abramovich and Chen. In the log Calabi-Yau case, if D is maximally degenerate, then we propose that Spec R is the mirror to X\D, while in the Calabi-Yau degeneration case, if the degeneration is maximally unipotent, the mirror is expected to be Proj R. The main result in this paper is that R as defined is an associative, commutative ring with unit, with associativity the most difficult part.