Calabi-Yau manifolds realizing symplectically rigid monodromy tuples
type: Item
instance of: scholarly article
title: Calabi-Yau manifolds realizing symplectically rigid monodromy tuples
zbMATH Open document ID: 1478.14025
author: C. F. Doran
author: Andreas Malmendier
published in: Advances in Theoretical and Mathematical Physics
publication date: 2021-11-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1503.07500
review text: This work is motivated by the study of mirror symmetry and Calabi-Yau manifolds, i.e., compact Kaehler manifolds with trivial canonical bundle. The \(K3\) surfaces are the only simply connected Calabi-Yau surfaces manifolds. The classification of Calabi-Yau threefolds remains an open problem. Systematic computer searches have given us a better idea of the landscape of Calabi-Yau threefolds (see [\textit{S. B. Johnson} and \textit{W. Taylor}, J. High Energy Phys. 2014, No. 10, Paper No. 023, 42 p. (2014; Zbl 1333.81384); \textit{W. Taylor}, J. High Energy Phys. 2012, No. 8, Paper No. 32, 18 p. (2012; Zbl 1397.14048)]). In particular they have showed the importance of elliptic fibrations on Calabi-Yau threefolds. Unfortunately, it has also been revealed that it is generally quite difficult to construct examples of families of Calabi-Yau threefolds with small Hodge number \(h^{2,1}\) by specialization of multi-parameter families. The mathematical study of mirror symmetry essentially began with the example of the quintic and mirror quintic family of \textit{Ph. Candelas} et al. [``An exactly soluble superconformal theory from a mirror pair of Calabi-Yau manifolds'', Phys. Lett. B 258, No. 1--2, 118--126 (1991; \url{doi:10.1016/0370-2693(91)91218-K})]. The quintic family is a generic quintic hypersurface in \(\mathbb P^5\), e.g., the Fermat quantic \(X^5_0+\cdot +X^5_4 = 0\) ; which is a Calabi-Yau threefold with Hodge numbers \(h^{1,1} = 1\) and \(h^{2,1} = 101\). This family is a one-dimensional family of Calabi-Yau threefolds defined over the base \(\mathbb P^1{0,1,\infty}\) and has exactly three singular fibers: the large complex structure limit, the Gepner point, and the stacky point. Moreover this family gives rise to a variation of Hodge structure and an associated Picard-Fuchs differential equation. In turn, the solutions to the differential equation determine this variation of Hodge structure. The Picard-Fuchs equations for other families of Calabi-Yau threefold…
reviewed by: Alessandro Ruzzi
Mathematics Subject Classification ID: 14D07
Mathematics Subject Classification ID: 14J32
Mathematics Subject Classification ID: 32G20
Mathematics Subject Classification ID: 14J33
zbMATH DE Number: 7431034
zbMATH Keywords: Calabi-Yau manifolds
zbMATH Keywords: elliptic surfaces
zbMATH Keywords: Picard-Fuchs equation
zbMATH Keywords: variation of Hodge structure
zbMATH Keywords: Euler integral transform
zbMATH Keywords: special functions
MaRDI profile type: Publication
arXiv ID: 1503.07500
DOI: 10.4310/ATMP.2019.V23.N5.A3
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description: scientific article