Lattice polarized toric K3 surfaces

TL;DR

This paper refines the lattice-based approach to mirror symmetry for toric K3 surfaces, resolving previous discrepancies and confirming mirror symmetry across all cases using explicit computer calculations.

Contribution

It provides a refined lattice framework that clarifies mirror symmetry for toric K3 hypersurfaces, including previously problematic cases.

Findings

Mirror symmetry holds for all toric K3 hypersurface families from dual reflexive polyhedra.

The refined lattice approach resolves discrepancies in Picard lattice ranks.

Explicit computer calculations confirm the theoretical predictions.

Abstract

When studying mirror symmetry in the context of K3 surfaces, the hyperkaehler structure of K3 makes the notion of exchanging Kaehler and complex moduli ambiguous. On the other hand, the metric is not renormalized due to the higher amount of supersymmetry of the underlying superconformal field theory. Thus one can define a natural mapping from the classical K3 moduli space to the moduli space of conformal field theories. Apart from the generalization of mirror constructions for Calabi-Yau threefolds, there is a formulation of mirror symmetry in terms of orthogonal lattices and global moduli space arguments. In many cases both approaches agree perfectly - with a long outstanding exception: Batyrev's mirror construction for K3 hypersurfaces in toric varieties does not fit into the lattice picture whenever the Picard group of the K3 surface is not generated by the pullbacks of the equivariant divisors of the ambient toric variety. In this case, not even the ranks of the corresponding Picard lattices add up as expected. In this paper the connection is clarified by refining the lattice picture. We show (by explicit calculation with a computer) mirror symmetry for all families of toric K3 hypersurfaces corresponding to dual reflexive polyhedra, including the formerly problematic cases.