Mirror symmetry and elliptic Calabi-Yau manifolds
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Springer
Received: November 28, 2018 Accepted: March 25, 2019 Published: April 10, 2019
Yu-Chien Huang and Washington Taylor Center for Theoretical Physics, Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, U.S.A.
E-mail: yc [email protected], [email protected] Abstract: We find that for many Calabi-Yau threefolds with elliptic or genus one fibrations mirror symmetry factorizes between the fiber and the base of the fibration. In the simplest examples, the generic CY elliptic fibration over any toric base surface B that supports an elliptic Calabi-Yau threefold has a mirror that is an elliptic fibration over a dual ˜ that is related through toric geometry to the line bundle −6KB . The toric base surface B Kreuzer-Skarke database includes all these examples and gives a wide range of other more complicated constructions where mirror symmetry also factorizes. Since recent evidence suggests that most Calabi-Yau threefolds are elliptic or genus one fibered, this points to a new way of understanding mirror symmetry that may apply to a large fraction of smooth Calabi-Yau threefolds. The factorization structure identified here can also apply for CalabiYau manifolds of higher dimension. Keywords: Differential and Algebraic Geometry, F-Theory, String Duality, Superstring Vacua ArXiv ePrint: 1811.04947
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP04(2019)083
JHEP04(2019)083
Mirror symmetry and elliptic Calabi-Yau manifolds
Contents 1 Introduction
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2 Toric hypersurface Calabi-Yau manifolds, fibrations and mirror symmetry 2.1 Toric hypersurfaces and fibrations 2.2 Factorization of mirror symmetry
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4 Some further examples 4.1 Tunings of generic fibrations (example: tuning an SU(2) on P2 ) ˜ 4.2 Tunings of generic fibrations over base B and mirror base B 4.3 Standard stacking F10 -fibered ∇ vs. non-standard F10 -fibered ∆ 4.4 Other toric fibers (example: vertex stacking on fiber F2 = P1 × P1 ) 4.5 Elliptic fibration with a non-fibered mirror 4.6 Elliptic Calabi-Yau fourfolds (example: generic elliptic fibration over P3 )
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5 Conclusions and further questions 5.1 Summary of results 5.2 Further questions and directions
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A The 16 reflexive 2D fiber polytopes ∇2
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B Faces of the base polytope and chains of non-Higgsable clusters
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Introduction
Calabi-Yau manifolds have been a major subject of study in mathematics and physics over the last three decades, following the realization that these geometries can be used to compactify string theory in a way that preserves supersymmetry [1]. One of the most intriguing aspects of Calabi-Yau threefolds is the existence of an equivalence known as mirror symmetry (see e.g. [2]) that relates the physics of a type IIA string compactification on a CalabiYau threefold X to that arising from a type IIB string compactification on a mirror threefold ˜ One of the first important clues to mirror symmetry was the observation [3, 4] that the X.
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