On the positivity and integrality of coefficients of mirror maps

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. Almost all 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.