The Monomial-Divisor Mirror Map

TL;DR

This paper explores the monomial-divisor mirror map, a natural construction linking Hodge groups of Calabi-Yau hypersurfaces in toric varieties, supporting mirror symmetry predictions and implications for moduli space connectivity.

Contribution

It introduces a natural construction of the monomial-divisor mirror map, clarifies its interpretation as a differential of the mirror isomorphism, and formulates a precise conjecture about the mirror isomorphism's form.

Findings

The monomial-divisor mirror map explicitly relates Hodge groups of mirror Calabi-Yau hypersurfaces.

The map can be viewed as the differential of the mirror isomorphism between moduli spaces.

Moduli spaces of different birational models are connected by analytic continuation, extending to other conformal field theories.

Abstract

For each family of Calabi-Yau hypersurfaces in toric varieties, Batyrev has proposed a possible mirror partner (which is also a family of Calabi-Yau hypersurfaces). We explain a natural construction of the isomorphism between certain Hodge groups of these hypersurfaces, as predicted by mirror symmetry, which we call the monomial-divisor mirror map. We indicate how this map can be interpreted as the differential of the expected mirror isomorphism between the moduli spaces of the two Calabi-Yau manifolds. We formulate a very precise conjecture about the form of that mirror isomorphism, which when combined with some earlier conjectures of the third author would completely specify it. We then conclude that the moduli spaces of the nonlinear sigma models whose targets are the different birational models of a Calabi-Yau space should be connected by analytic continuation, and that further analytic continuation should lead to moduli spaces of other kinds of conformal field theories. (This last conclusion was first drawn by Witten.)