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Supercongruences for rigid hypergeometric Calabi–Yau threefolds
Advances in Mathematics. 2021. Vol. 393. Article 108058.
Zudilin W., Long L.
We establish the supercongruences for the fourteen rigid hypergeometric Calabi-Yau threefolds over Q conjectured by Rodriguez-Villegas in 2003. Our first method is based on Dwork's theory of p-adic unit roots and it allows us to establish the supercongruences between the truncated hypergeometric series and the corresponding unit roots for ordinary primes. The other method makes use of the theory of hypergeometric motives, in particular, adapts the techniques from the recent work of Beukers, Cohen and Mellit on finite hypergeometric sums over Q. Essential ingredients in executing the both approaches are the modularity of the underlying Calabi-Yau threefolds and a p-adic perturbation method applied to hypergeometric functions. (c) 2021 Elsevier Inc. All rights reserved.
Keywords: mirror symmetry
Publication based on the results of:
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Basalaev A., Journal of Geometry and Physics 2025 Vol. 215 Article 105538
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Varolgunes U., Polishchuk A., Mathematische Annalen 2024 Vol. 388 P. 2331–2386
We consider Takahashi’s categorical interpretation of the Berglund–Hubsch mirror symmetry conjecture for invertible polynomials in the case of chain polynomials. Our strategy is based on a stronger claim that the relevant categories satisfy a recursion of directed -categories, which may be of independent interest. We give a full proof of this claim on the B-side. On ...
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A nef-partition for a weighted complete intersection is a combinatorial structure on its weights and degrees which is important for Mirror Symmetry. It is known that nef-partitions exist for smooth well-formed Fano weighted complete intersections of small dimension or codimension, and that in these cases they are strong in the sense that they can be ...
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Horja R. P., Katzarkov Ludmil, Advances in Mathematics 2024 Vol. 453 Article 109831
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We study Landau-Ginzburg orbifolds (f,G) with f=xn1+…+xnN and G=S⋉Gd, where S⊆SN and Gd is either the maximal group of scalar symmetries of f or the intersection of the maximal diagonal symmetries of f with SLN(ℂ). We construct a mirror map between the corresponding phase spaces and prove that it is an isomorphism restricted to a ...
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Chiodo A., Kalashnikov E. G., Veniani D. C., Advances in Mathematics: Scientific Journal 2020 Vol. 363
We generalize the cohomological mirror duality of Borcea and Voisin in any dimension and for any number of factors. Our proof applies to all examples which can be constructed through Berglund–Hübsch duality. Our method is a variant of the so-called Landau–Ginzburg/Calabi–Yau correspondence of Calabi–Yau orbifolds with an involution that does not preserve the volume form. ...
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Gusein-Zade S., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2020 Vol. 16 No. 051 P. 1–15
P. Berglund, T. Hübsch, and M. Henningson proposed a method to construct mirror symmetric Calabi–Yau manifolds. They considered a pair consisting of an invertible polynomial and of a finite (abelian) group
of its diagonal symmetries together with a dual pair. A. Takahashi suggested a method to generalize this construction to symmetry groups generated by some diagonal ...
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Buryak A., Moscow Mathematical Journal 2020 Vol. 20 No. 3 P. 475–493
By a famous result of K. Saito, the parameter space of the miniversal deformation of the $A_{r-1}$-singularity carries a Frobenius manifold structure. The Landau-Ginzburg mirror symmetry says that, in the flat coordinates, the potential of this Frobenius manifold is equal to the generating series of certain integrals over the moduli space of $r$-spin curves. In ...
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We verify Katzarkov-Kontsevich-Pantev conjecture for Landau-Ginzburg models of smooth Fano threefolds. ...
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Barannikov S., / Series HAL "math". 2018. No. 01804639.
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Sawada T., Zaidi Q., Journal of Mathematical Psychology 2018 Vol. 87 P. 108–125
A 3D shape of an object is N-fold rotational-symmetric if the shape is invariant for 360/N degree rotations about an axis. Human observers are sensitive to the 2D rotational-symmetry of a retinal image, but they are less sensitive than they are to 2D mirror-symmetry, which involves invariance to reflection across an axis. Note that perception of the ...
Added: October 1, 2018
Jayadevan V., Sawada T., Delp E. et al., Symmetry 2018 Vol. 10 No. 8: 344 P. 1–24
This is an empirical study with a computational model.
The human visual system uses priors to convert an ill-posed inverse problem of 3D shape recovery into a well-posed one. In previous studies, we have demonstrated the use of priors like symmetry, compactness and minimal surface in the perception of 3D symmetric shapes. We also showed that ...
Added: September 3, 2018
Galkin S., Iritani H., , in: Primitive Forms and Related Subjects — Kavli IPMU 2014.: Tokyo: Mathematical Society of Japan, 2019. P. 55–115.
The asymptotic behaviour of solutions to the quantum differential equation of a Fano manifold F defines a characteristic class A_F of F, called the principal asymptotic class.
Gamma conjecture of Vasily Golyshev and the present authors claims that the principal asymptotic class A_F equals the Gamma class associated to Euler's Gamma-function.
We illustrate in the case of ...
Added: September 1, 2018
Coates T., Galkin S., Kasprzyk A. et al., Experimental Mathematics 2020 Vol. 29 No. 2 P. 183–221
We collect a list of known four-dimensional Fano manifolds and compute their quantum periods. This list includes all four-dimensional Fano manifolds of index greater than one, all four-dimensional toric Fano manifolds, all four-dimensional products of lower-dimensional Fano manifolds, and certain complete intersections in projective bundles. ...
Added: September 1, 2018
Kalinin N., Guzmán-Sáenz A., Prieto Y. et al., Proceedings of the National Academy of Sciences of the United States of America 2018 Vol. 115 No. 35 P. E8135–E8142
Tropical geometry, an established field in pure mathematics, is a place where string theory, mirror symmetry, computational algebra, auction theory, and so forth meet and influence one another. In this paper, we report on our discovery of a tropical model with self-organized criticality (SOC) behavior. Our model is continuous, in contrast to all known models ...
Added: August 28, 2018