On the positivity and integrality of coefficients of mirror maps

TL;DR

This paper explores the positivity and integrality properties of mirror maps in mirror symmetry, proposing conjectures that the naive mirror map always has positive integer coefficients, while the true mirror map has integer coefficients.

Contribution

It introduces conjectural generalizations of mirror map properties, distinguishing between true and naive mirror maps, and provides evidence and conditions under which their coefficients are integral or positive.

Findings

Naive mirror maps are conjectured to always have positive integer coefficients.

True mirror maps are conjectured to have integer coefficients, not necessarily positive.

The paper provides computational evidence and theoretical conditions related to mirror map coefficients.

Abstract

We present natural conjectural generalizations of the `positivity and integrality of mirror maps' phenomenon, encompassing the mirror maps appearing in the Batyrev--Borisov construction of mirror Calabi--Yau complete intersections in Fano toric varieties as a special case. We find that, given the combinatorial data from which one constructs a mirror pair of Calabi--Yau complete intersections, there are two ways of writing down an associated `mirror map': one which is the `true mirror map', meaning the one which appears in mirror symmetry theorems; and one which is the `naive mirror map'. The two are equal under a certain combinatorial criterion which holds e.g. for the quintic threefold, but not in general. We conjecture (based on substantial computer checks, together with proofs under extra hypotheses) that the naive mirror map always has positive integer coefficients, while the true mirror map always has integer (but not necessarily positive) coefficients. Most previous works on the integrality of mirror maps concern the naive mirror map, and in particular, only apply to the true mirror map under the combinatorial criterion mentioned above.