Canonical Coordinates in Toric Degenerations

TL;DR

This paper proves the triviality of the mirror map for canonical Calabi-Yau families, showing that their natural coordinates are canonical in Hodge theory, which simplifies the extraction of enumerative information from periods.

Contribution

It establishes the triviality of the mirror map in canonical Calabi-Yau families and explicitly computes period integrals using tropical geometry techniques.

Findings

Mirror map is trivial for these families

Canonical coordinates align with Hodge-theoretic notions

Explicit period integrals are computed using tropical cycles

Abstract

We prove that the mirror map is trivial for the canonical formal families of Calabi-Yau varieties constructed by Gross and the second author. In other words, the natural coordinate in a canonical Calabi-Yau family is a canonical coordinate in the sense of Hodge theory. This implies that the higher weight periods directly carry enumerative information with no further gauging necessary as opposed to the classical case. A side result is that the canonical formal families lift to analytic families. We compute the relevant period integrals explicitly. The cycles to integrate over are constructed from tropical 1-cycles in the intersection complex of the degenerate Calabi-Yau.