The arithmetic mirror symmetry and Calabi-Yau manifolds
type: Item
instance of: scholarly article
title: The arithmetic mirror symmetry and Calabi-Yau manifolds
zbMATH Open document ID: 0962.14028
published in: Communications in Mathematical Physics
publication date: 2001-06-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/alg-geom/9612002
review text: In this paper the authors try to find a relation between their version of mirror symmetry for K3 surfaces [\textit{V. A. Gritsenko} and \textit{V. V. Nikulin}, Math. Res. Let. 3, No. 2, 211-229 (1996; Zbl 0871.14031)] and mirror symmetry for Calabi-Yau manifolds fibered in K3 surfaces. The difference between the authors' version of mirror symmetry for K3's and the version of \textit{I. V. Dolgachev} [J. Math. Sci., New York 81, No. 3, 2599-2630 (1996; Zbl 0890.14024)] is that they take into account the existence of an automorphic form on the \(B\)-model side which is conjectured to fulfill a Borcherds type identity. In this way they reflect information about effective curves on K3 surfaces by properties of functions on the moduli space of the mirror family. In the paper under review a few examples are mentioned where such automorphic forms are known to exist. The authors formulate the problem to find examples and to classify Calabi-Yau 3-folds fibered in K3 surfaces whose Picard or transcendental lattice has a certain property (2-reflective). This short paper does not contain any proof. At the end the authors present a very vague formulation of a conjecture relating the automorphic forms considered above to the Yukawa coupling and quantum cohomology of a Calabi-Yau manifold fibered in K3 surfaces.
reviewed by: Bernd Kreußler
Mathematics Subject Classification ID: 14J32
Mathematics Subject Classification ID: 81T30
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 32N10
Mathematics Subject Classification ID: 32Q25
zbMATH DE Number: 1445810
zbMATH Keywords: Kac-Moody algebra
zbMATH Keywords: Lorentzian
zbMATH Keywords: symmetric domain
zbMATH Keywords: automorphic form
zbMATH Keywords: Calabi-Yau manifold
zbMATH Keywords: mirror symmetry
zbMATH Keywords: K3 surface
zbMATH Keywords: hyperbolic lattice
zbMATH Keywords: reflection group
zbMATH Keywords: Yukawa coupling
zbMATH Keywords: quantum cohomology
author: V. Gritsenko
author: V. V. Nikulin
MaRDI profile type: Publication
OpenAlex ID: W1747975156
arXiv ID: alg-geom/9612002
DOI: 10.1007/S002200050769
description: scientific article