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.
- Publication:
-
arXiv e-prints
- Pub Date:
- April 2017
- DOI:
- 10.48550/arXiv.1704.00164
- arXiv:
- arXiv:1704.00164
- Bibcode:
- 2017arXiv170400164V
- Keywords:
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- Mathematics - Algebraic Geometry;
- 14J32;
- 32S40
- E-Print:
- 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