A lecture on Arithmetic Mirror Symmetry and Calabi-Yau manifolds

Abstract

We extend our variant of mirror symmetry for K3 surfaces \cite{GN3} and clarify its relation with mirror symmetry for Calabi-Yau manifolds. We introduce two classes (for the models A and B) of Calabi-Yau manifolds fibrated by K3 surfaces with some special Picard lattices. These two classes are related with automorphic forms on IV type domains which we studied in our papers \cite{GN1}-\cite{GN6}. Conjecturally these automorphic forms take part in the quantum intersection pairing for model A, Yukawa coupling for model B and mirror symmetry between these two classes of Calabi-Yau manifolds. Recently there were several papers by physicists where it was shown on some examples. We propose a problem of classification of introduced Calabi-Yau manifolds. Our papers \cite{GN1}-\cite{GN6} and \cite{N3}-\cite{N14} give a hope that this is possible. They describe possible Picard or transcendental lattices of general K3 fibers of the Calabi-Yau manifolds.

AMS-Tex, 11 pages, no figures. The variant prepared for publication; many small changes introduced

References in corpus (7)

• Algebras, BPS States, and Strings
• Counting Dyons in N=4 String Theory
• Equivariant Gromov - Witten Invariants
• The Igusa modular forms and ``the simplest'' Lorentzian Kac--Moody algebras
• The Remark on Discriminants of K3 Surfaces Moduli as Sets of Zeros of Automorphic Forms
• Counting Curves with Modular Forms
• K3 surfaces with interesting groups of automorphisms

Cited by in corpus (11)

• The Siegel modular forms of genus 2 with the simplest divisor
• Lorentzian Kac-Moody algebras with Weyl groups of 2-reflections
• Reflective modular forms in algebraic geometry
• Umbral Moonshine and the Niemeier Lattices
• The Singular Theta Correspondence, Lorentzian Lattices and Borcherds-Kac-Moody Algebras
• On the classification of hyperbolic root systems of the rank three. Part III
• Modular forms of orthogonal type and Jacobi theta-series
• Uniruledness of orthogonal modular varieties
• A Lecture about classification of Lorentzian Kac-Moody algebras of the rank three
• Reflective modular forms: A Jacobi forms approach
• Examples of lattice-polarized K3 surfaces with automorphic discriminant, and Lorentzian Kac--Moody algebras