Calabi--Yau Operators

Abstract

Motivated by mirror symmetry of one-parameter models, an interesting class of Fuchsian differential operators can be singled out, the so-called Calabi--Yau operators, introduced by Almkvist and Zudilin. They conjecturally determine $Sp(4)$-local systems that underly a $\mathbb{Q}$-VHS with Hodge numbers \[h^{3 0}=h^{2 1}=h^{1 2}=h^{0 3}=1\] and in the best cases they make their appearance as Picard--Fuchs operators of families of Calabi--Yau threefolds with $h^{12}=1$ and encode the numbers of rational curves on a mirror manifold with $h^{11}=1$. We review some of the striking properties of this rich class of operators.

This paper of expository character is an extended written version of a talk given at the conference "Uniformization, Riemann-Hilbert Correspondence, Calabi-Yau Manifolds, and Picard-Fuchs Equations" held 13.-18.07.2015 at the Mittag-Leffler Institute which was organised by L. Ji and S.-T. Yau and will appear in the conference proceedings

References in corpus (4)

• Mirror Symmetry and Integral Variations of Hodge Structure Underlying One Parameter Families of Calabi-Yau Threefolds
• Multiple binomial sums
• Calabi-Yau manifolds realizing symplectically rigid monodromy tuples
• Cluster duality and mirror symmetry for Grassmannians

Cited by in corpus (3)

• Special Geometry and the Swampland
• Movable vs Monodromy Nilpotent Cones of Calabi-Yau Manifolds
• Diagonals of rational functions: from differential algebra to effective algebraic geometry (unabridged version)