Abstract
We consider the phenomenon of the complete coincidence of key properties of Calabi–Yau manifolds realized as hypersurfaces in two different weighted projective spaces. More precisely, the first manifold in such a pair is realized as a hypersurface in a weighted projective space, and the second is realized as a hypersurface in an orbifold of another weighted projective space. The two manifolds in each pair have the same Hodge numbers and the same geometry on the complex structure moduli space and are also associated with the same \(N{=}2\) gauged linear sigma model. We explain these coincidences using the correspondence between Calabi–Yau manifolds and the Batyrev reflexive polyhedra.
Similar content being viewed by others
References
P. Berglund and T. Hübsch, “A generalized construction of mirror manifolds,” Nucl. Phys. B, 393, 377–391 (1993); arXiv:hep-th/9201014v1 (1992).
P. Berglund and T. Hübsch, “A generalized construction of Calabi–Yau models and mirror symmetry,” SciPost Phys., 4, 009 (2018); arXiv:1611.10300v3 [hep-th] (2016).
H. Jockers, V. Kumar, J. M. Lapan, D. R. Morrison, and M. Romo, “Two-sphere partition functions and Gromov–Witten invariants,” Commun. Math. Phys., 325, 1139–1170 (2014); arXiv:1208.6244v3 [hep-th] (2012).
E. Witten, “Phases of \(N{=}2\) theories in two-dimensions,” Nucl. Phys. B, 403, 159–222 (1993); arXiv:hep-th/9301042v3 (1993).
K. Aleshkin and A. Belavin, “A new approach for computing the geometry of the moduli spaces for a Calabi–Yau manifold,” J. Phys. A: Math. Theor., 51, 055403 (2018); arXiv:1706.05342v4 [hep-th] (2017).
K. Aleshkin and A. Belavin, “Special geometry on the 101 dimesional moduli space of the quintic threefold,” JHEP, 1803, 018 (2018); arXiv:1710.11609v3 [hep-th] (2017).
K. Aleshkin and A. Belavin, “Exact computation of the special geometry for Calabi–Yau hypersurfaces of Fermat type,” JETP Lett., 108, 705–709 (2018); arXiv:1806.02772v2 [hep-th] (2018).
K. Aleshkin, A. Belavin, and A. Litvinov, “Two-sphere partition functions and Kahler potentials on CY moduli spaces,” JETP Lett., 108, 710 (2018).
K. Aleshkin, A. Belavin, and A. Litvinov, “JKLMR conjecture and Batyrev construction,” J. Stat. Mech., 2019, 034003 (2019); arXiv:1812.00478v3 [hep-th] (2018).
V. V. Batyrev, “Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties,” J. Alg.Geom., 3, 493–535 (1994); arXiv:alg-geom/9310003v1 (1993).
W. Lerche, C. Vafa, and N. P. Warner, “Chiral rings in \(N{=}2\) superconformal theories,” Nucl. Phys., 324, 427–474 (1989).
P. Candelas and X. C. de la Ossa, “Moduli space of Calabi–Yau manifolds,” Nucl. Phys. B, 355, 455–481 (1991).
P. Candelas, X. C. De La Ossa, P. S. Green, and L. Parkes, “A pair of Calabi–Yau manifolds as an exactly soluble superconformal theory,” Nucl. Phys. B, 359, 21–74 (1991).
P. Berglund, P. Candelas, X. C. de la Ossa, A. Font, T. Hübsch, D. Jančić, and F. Quevedo, “Periods for Calabi–Yau and Landau–Ginzburg vacua,” Nucl. Phys. B, 419, 352–403 (1994); arXiv:hep-th/9308005v2 (1993).
A. Chiodo, H. Iritani, and Y. Ruan, “Landau–Ginzburg/Calabi–Yau correspondence: Global mirror symmetry and Orlov equivalence,” Publ. IHES, 119, 127–216 (2013); arXiv:1201.0813v3 [math.AG] (2012).
K. Aleshkin and A. Belavin, “Special geometry on the moduli space for the two-moduli non-Fermat Calabi–Yau,” Phys. Lett. B, 776, 139–144 (2018); arXiv:1708.08362v2 [hep-th] (2017).
G. Bonelli, A. Sciarappa, A. Tanzini, and P. Vasko, “Vortex partition functions, wall crossing, and equivariant Gromov–Witten invariants,” Commun. Math. Phys., 333, 717–760 (2015); arXiv:1307.5997v2 [hep-th] (2013).
J. Gomis and S. Lee, “Exact Kähler potential from gauge theory and mirror symmetry,” JHEP, 1304, 019 (2013).
N. Doroud and J. Gomis, “Gauge theory dynamics and Kähler potential for Calabi–Yau complex moduli,” JHEP, 1312, 099 (2013); arXiv:1309.2305v2 [hep-th] (2013).
E. Gerchkovitz, J. Gomis, and Z. Komargodski, “Sphere partition functions and the Zamolodchikov metric,” JHEP, 1411, 001 (2014); arXiv:1405.7271v2 [hep-th] (2014).
F. Benini and S. Cremonesi, “Partition functions of \(\mathcal{N}{=}(2,2)\) gauge theories on \(\text{S}^2\) and vortices,” Commun. Math. Phys., 334, 1483–1527 (2015); arXiv:1206.2356v3 [hep-th] (2012).
N. Doroud, J. Gomis, B. Le Floch, and S. Lee, “Exact results in \(D{=}2\) supersymmetric gauge theories,” JHEP, 1305, 093 (2013); arXiv:1206.2606v4 [hep-th] (2012).
K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow, “Chap. 7: Toric geometry for string theory,” in: Mirror Symmetry (Clay Math. Monogr., Vol. 1), Amer. Math. Soc., Providence, R. I. (2003), pp. 101–142; arXiv:hep-th/0002222v3 (2000).
A. A. Belavin and B. A. Eremin, “Partition functions of \(\mathcal{N}=(2,2)\) supersymmetric sigma models and special geometry on the moduli spaces of Calabi–Yau manifolds,” Theor. Math. Phys., 201, 1606–1613 (2019).
P. Candelas, X. C. de la Ossa, and S. Katz, “Mirror symmetry for Calabi–Yau hypersurfaces in weighted and extensions of Landau–Ginzburg theory,” Nucl. Phys. B, 450, 267–290 (1995); arXiv:hep-th/9412117v1 (1994).
D. Favero and T. L. Kelly, “Derived categories of BHK mirrors,” Adv. Math., 352, 943–980 (2019); arXiv:1602.05876v2 [math.AG] (2016).
Acknowledgments
The authors are grateful to G. Koshevoy and A. Litvinov for the useful discussions.
Funding
This research was supported by a grant from the Russian Science Foundation (Project No. 18-12-00439).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The authors declare no conflicts of interest.
Rights and permissions
About this article
Cite this article
Belavin, A.A., Belakovskii, M.Y. Coincidences between Calabi–Yau manifolds of Berglund–Hübsch type and Batyrev polytopes. Theor Math Phys 205, 1439–1455 (2020). https://doi.org/10.1134/S0040577920110045
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577920110045