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## Mirror map for Fermat polynomials with a nonabelian group of symmetries

Theoretical and Mathematical Physics. 2021. Vol. 209. No. 2. P. 1491-1506.

Basalaev A., Ionov A.

We study Landau-Ginzburg orbifolds (*f*,*G*) with *f*=*x**n*1+…+*x**n**N* and *G*=*S*⋉*G**d*, where *S*⊆*S**N* and *G**d* is either the maximal group of scalar symmetries of *f* or the intersection of the maximal diagonal symmetries of *f* with SL*N*(ℂ). We construct a mirror map between the corresponding phase spaces and prove that it is an isomorphism restricted to a certain subspace of the phase space when *n*=*N* is a prime number. When *S* satisfies the condition PC of Ebeling and Gusein-Zade this subspace coincides with the full space. We also show that two phase spaces are isomorphic for *n*=*N*=5.

Akhtar M., Coates T., Galkin S. et al., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2012 Vol. 8 No. 094 P. 1-707

Given a Laurent polynomial f, one can form the period of f: this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds. Mutations are a particular class of birational transformations acting on Laurent polynomials in two variables; they preserve the period and are closely connected with ...

Added: September 14, 2013

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

Cruz Morales J. A., Galkin S., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2013 Vol. 9 No. 005 P. 1-13

In this note we provide a new, algebraic proof of the excessive Laurent phenomenon for mutations of potentials (in the sense of [Galkin S., Usnich A., Preprint IPMU 10-0100, 2010]) by introducing to this theory the analogue of the upper bounds from [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1–52]. ...

Added: May 27, 2013

Nakatsu T., Kato A., Noumi M. et al., Physics Letters B 1994 Vol. 322 No. 3 P. 192-197

We study the relation between topological string theory and singularity theory using the partition function of A_N-1 topological string defined by matrix integral of Kontsevich type. Genus expansion of the free energy is considered, and the genus g=0 contribution is shown to be described by a special solution of N-reduced dispersionless KP system. We show ...

Added: August 14, 2014

Galkin S., Golyshev V., Iritani H., Duke Mathematical Journal 2016 Vol. 165 No. 11 P. 2005-2077

We propose Gamma Conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class A_F to a Fano manifold F. We say that F satisfies Gamma Conjecture I if A_F equals the ...

Added: November 18, 2014

Coates T., Corti A., Galkin S. et al., Geometry and Topology 2016 Vol. 20 No. 1 P. 103-256

The quantum period of a variety X is a generating function for certain Gromov-Witten invariants of X which plays an important role in mirror symmetry. In this paper we compute the quantum periods of all 3-dimensional Fano manifolds. In particular we show that 3-dimensional Fano manifolds with very ample anticanonical bundle have mirrors given by ...

Added: November 18, 2014

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 ...

Added: October 27, 2020

Gritsenko V., Никулин В. В., TRANSACTIONS OF THE MOSCOW MATHEMATICAL SOCIETY 2017 Т. 78 № 1 С. 89-100

Using our results about Lorentzian Kac--Moody algebras and arithmetic mirror symmetry, we give six series of examples of lattice-polarized K3 surfaces with automorphic discriminant. ...

Added: October 11, 2017

Galkin S., Belmans P., Mukhopadhyay S., Graph potentials and moduli spaces of rank two bundles on a curve / Cornell University. Series math "arxiv.org". 2020. No. 2009.05568.

We introduce graph potentials, which are Laurent polynomials associated to (colored) trivalent graphs. These graphs encode degenerations of curves to rational curves, and graph potentials encode degenerations of the moduli space of rank 2 bundles with fixed determinant. We show that the birational type of the graph potential only depends on the homotopy type of ...

Added: April 15, 2021

Coates T., Galkin S., Kasprzyk A. et al., Quantum Periods For Certain Four-Dimensional Fano Manifolds / Cornell University. Series math "arxiv.org". 2014. No. 1406.4891.

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: June 20, 2014

Galkin S., Rybakov S., A family of K3 surfaces and towers of algebraic curves over finite fields / Cornell University. Series math "arxiv.org". 2019. No. 1910.14379.

For a family of K3 surfaces we implement a variation of a general construction of towers of algebraic curves over finite fields given in a previous paper. As a result we get a good tower over k=𝔽_{p^2}, that is optimal if p=3. ...

Added: November 6, 2019

Galkin S., Iritani H., Gamma conjecture via mirror symmetry / Cornell University. Series math "arxiv.org". 2015. No. 1508.00719.

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 G_F associated to Euler's Γ-function. We illustrate in ...

Added: August 5, 2015

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

Coates T., Corti A., Galkin S. et al., Mirror Symmetry and Fano Manifolds / Cornell University. Series math "arxiv.org". 2012. No. 1212.1722.

We consider mirror symmetry for Fano manifolds, and describe how one can recover the classification of 3-dimensional Fano manifolds from the study of their mirrors. We sketch a program to classify 4-dimensional Fano manifolds using these ideas. ...

Added: September 14, 2013

Ebeling W., Gusein-Zade S., International Mathematics Research Notices 2021 Vol. 2021 No. 16 P. 12305-12329

A.Takahashi suggested a conjectural method to find mirror symmetric pairs consisting of invertible polynomials and symmetry groups generated by some diagonal symmetries and some permutations of variables. Here we generalize the Saito duality between Burnside rings to a case of non-abelian groups and prove a "non-abelian" generalization of the statement about the equivariant Saito duality ...

Added: August 26, 2021

Ionov A., Supersymmetric Semisimple Cardy-Frobenius Algebras / . 2015.

Cardy-Frobenius algebra is the algebraic structure on the space of states in open-closed topological field theory. We prove that every semisimple super Cardy-Frobenius algebras is the direct sum of the super Cardy-Frobenius algebras of three simple types. We also apply our results to singularity theory via Landau-Ginzburg models and matrix factorizations. ...

Added: November 8, 2016

Ionov A., Primitive forms for Gepner singularities / Cornell University. Series arXiv "math". 2016. No. 1611.03962.

We apply the technique of the paper "The abelian/nonabelian correspondence and Frobenius manifolds" by I. Ciocan-Fontanine, B. Kim, C. Sabbah to construct Saito primitive forms for Gepner singularities. ...

Added: November 16, 2016

Ionov A., Journal of Geometry and Physics 2019 Vol. 140 P. 125-130

We provide a construction of Saito primitive forms for Gepner singularity by studying the relation between Saito primitive forms for Gepner singularities and primitive forms for singularities of the form F_{k,n} = ∑^n_{i=1} x^k_i invariant under the natural S_n-action. ...

Added: November 8, 2019

Sawada T., Li Y., Pizlo Z., Symmetry 2011 Vol. 3 No. 2 P. 365-388

Added: September 23, 2014

Ilten N. O., Lewis J., Przyjalkowski V., Journal of Algebra 2013 Vol. 374 P. 104-121

We show that every Picard rank one smooth Fano threefold has a weak Landau–Ginzburg model coming from a toric degeneration. The fibers of these Landau–Ginzburg models can be compactified to K3 surfaces with Picard lattice of rank 19. We also show that any smooth Fano variety of arbitrary dimension which is a complete intersection of ...

Added: July 2, 2013

Coates T., Corti A., Galkin S. et al., , in: European Congress of Mathematics Kraków, 2 – 7 July, 2012. .: Zürich: European Mathematical Society Publishing house, 2014.. Ch. 16. P. 285-300.

We consider mirror symmetry for Fano manifolds, and describe how one can recover the classification of 3-dimensional Fano manifolds from the study of their mirrors. We sketch a program to classify 4-dimensional Fano manifolds using these ideas. ...

Added: February 19, 2014

Vassiliev V., Moscow Mathematical Journal 2017 Vol. 17 No. 4 P. 825-836

The local multiplicities of the caustics, the Maxwell sets, and the complex Stokes' sets in the spaces of versal deformations of Pham singularities (that is, of germs of holomorphic functions C^n --> C^1 which are expressed by the sums of degrees of the coordinate functions) are calculated ...

Added: December 27, 2017

Galkin S., The conifold point / Cornell University. Series math "arxiv.org". 2014. No. 1404.7388.

Consider a Laurent polynomial with real positive coefficients such that the origin is strictly inside its Newton polytope. Then it is strongly convex as a function of real positive argument. So it has a distinguished Morse critical point --- the unique critical point with real positive coordinates. As a consequence we obtain a positive answer ...

Added: May 4, 2014

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