Lattice polarized toric K3 surfaces

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.

23 pages, Latex. For associated data, see http://www.th.physik.uni-bonn.de/People/rohsiepe/k3

References in corpus (4)

• Phases of $N=2$ Theories In Two Dimensions
• K3 Surfaces and String Duality
• Towards the Mirror Symmetry for Calabi-Yau Complete intersections in Gorenstein Toric Fano Varieties
• Orbifold Constructions of K3: A Link between Conformal Field Theory and Geometry

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• Mirror Symmetry for $G_2$-Manifolds: Twisted Connected Sums and Dual Tops
• Picard group of hypersurfaces in toric 3-folds
• Families of lattice polarized K3 surfaces with monodromy
• Period- and mirror-maps for the quartic K3
• The 14th case VHS via K3 fibrations
• Brill-Noether-general Limit Root Bundles: Absence of vector-like Exotics in F-theory Standard Models
• Fibration structures in toric Calabi-Yau Fourfolds
• Mirror pairs of Calabi-Yau threefolds from mirror pairs of quasi-Fano threefolds
• Mirror symmetry and K3 surfaces
• Short Tops and Semistable Degenerations
• Root bundles: Applications to F-theory Standard Models
• Vafa-Witten Theory: Invariants, Floer Homologies, Higgs Bundles, a Geometric Langlands Correspondence, and Categorification (String Math 2022 Proceedings)
• On mirrors of elliptically fibered K3 surfaces