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## Wess-Zumino-Witten model on elliptic curves at the critical level.

Journal of Physics A: Mathematical and Theoretical. 2001. Vol. 34. No. 11. P. 2403-2413.

Takebe T., Kuroki G.

We construct a Gaudin type lattice model as the Wess-Zumino-Witten model on elliptic curves at the critical level. Bethe eigenvectors are obtained by the bosonisation technique.

Takebe T., International Journal of Modern Physics A 2004 Vol. 19, May suppl. P. 418-435

Trigonometric degeneration of the Baxter-Belavin elliptic r matrix is described by the degeneration of the twisted WZW model on elliptic curves. The spaces of conformal blocks and conformal coinvariants of the degenerate model are factorised into those of the orbifold WZW model. ...

Added: August 14, 2014

Rybakov S., Mathematical notes 2016 Vol. 99 No. 3 P. 397-405

Let S be a bielliptic surface over a finite field, and let an elliptic curve B be the Albanese variety of S; then the zeta function of the surface S is equal to the zeta function of the direct product P1 × B. Therefore, the classification problem for the zeta functions of bielliptic surfaces is ...

Added: July 8, 2016

Brown F., / arxive. Series math "nt". 2013. No. 1110.6917.

Abstract. We study the de Rham fundamental group of the configuration space
E (n) of n + 1 marked points on a complex elliptic curve E, and define multiple
elliptic polylogarithms. These are multivalued functions on E (n) with unipotent
monodromy, and are constructed by a general averaging procedure. We show
that all iterated integrals on E (n) , ...

Added: May 14, 2014

Lebedev P. A., Nesterenko A., Черновик статьи 2014

В работе предложен алгоритм вычисления явного вида эндоморфизма эллиптической кривой, который может использоваться для ускорения операций по вычислению кратной точки. Приведены детали авторской реализации и результаты её производительности. В статье впервые представлены решения поставленной задачи для степени соответствующего многочлена Гильберта выше пятой. ...

Added: October 23, 2014

Matveeva A., Poberezhny V. A., Математические заметки 2017 Т. 101 № 1 С. 91-100

A one-dimensional generalization of the Riemann–Hilbert problem from the Riemann sphere to an elliptic curve is considered. A criterion for its positive solvability is obtained and an explicit form of all possible solutions is found. As in the spherical case, the solutions turn out to be isomonodromic. ...

Added: October 18, 2016

Lvovsky S., / Cornell University. Series arXiv "math". 2018.

We show that if we are given a smooth non-isotrivial family of elliptic curves over ℂ with a smooth base B for which the general fiber of the mapping J:B→𝔸^1 (assigning j-invariant of the fiber to a point) is connected, then the monodromy group of the family (acting on H1(⋅,ℤ) of the fibers) coincides with SL(2,ℤ); if the general fiber has m≥2 connected components, then the ...

Added: December 5, 2018

A.V.Zabrodin, Zotov A. V., Liashyk A. et al., Theoretical and Mathematical Physics 2017 Vol. 192 No. 2 P. 1141-1153

We discuss the correspondence between models solved by the Bethe ansatz and classical integrable systems of the Calogero type. We illustrate the correspondence by the simplest example of the inhomogeneous asymmetric six-vertex model parameterized by trigonometric(hyperbolic) functions. ...

Added: October 26, 2017

Poberezhny V. A., Matveeva A., Journal of Geometry and Physics 2017 Vol. 114 P. 384-393

We consider a generalization of Riemann–Hilbert problem on elliptic curves. For a given elliptic curve and irreducible representation of free group with two generators we construct explicitly a semistable vector bundle of degree zero obeying a logarithmic connection such that its monodromy over fundamental parallelogram is equivalent to given free group representation, monodromy along a−cycle is ...

Added: October 26, 2016

Brown F., / arxive. Series math "nt". 2013. No. arXiv:1110.6917v2.

Abstract. We study the de Rham fundamental group of the configuration space
E (n) of n + 1 marked points on a complex elliptic curve E, and define multiple
elliptic polylogarithms. These are multivalued functions on E (n) with unipotent
monodromy, and are constructed by a general averaging procedure. We show
that all iterated integrals on E (n) , ...

Added: May 14, 2014

Brown F., Levin A., / Cornell University. Series arXiv "math". 2013. No. 1110.6917 [.

Abstract. We study the de Rham fundamental group of the configuration sp ace of several marked points on a complex elliptic curve, and define multiple elliptic polylogarithms. These are multivalued functions with unipotent monodromy, and are constructed by a general averaging proce dure. We show that all iterated integrals on this configuration space can be ...

Added: October 4, 2013

Poberezhny V. A., / ИТЭФ. Series "Препринты ИТЭФ". 2012. No. 57/12.

In this work we investigate the action of generalized Schlesinger transformation on the isomonodromic families of meromorphic connections on the linear bundles of rank two and degree zero over an elliptic curve. The main interest is the action of the gauge transformation on the moduli space of vector bundles. the central result is the explicit ...

Added: March 31, 2014

Pavel Pyatov, Povolotsky A. M., Rittenberg V., Journal of Statistical Mechanics: Theory and Experiment 2018 Vol. 2018 No. 053107 P. 1-26

We study the large deviation functions for two quantities characterizing the avalanche dynamics in the Raise and Peel model: the number of tiles removed by avalanches and the number of global avalanches extending through the whole system. To this end, we exploit their connection to the groundstate eigenvalue of the XXZ model with twisted boundary ...

Added: July 17, 2018

Netay I. V., Савватеев А. В., / Cornell University. Series math "arxiv.org". 2016.

The paper is devoted to the description of family of scalene triangles for which the triangle formed by the intersection points of bisectors with opposite sides is isosceles.
We call them Sharygin triangles.
It turns out that they are parametrized by an open subset of an elliptic curve.
Also we prove that there are infinitely many non-similar integer ...

Added: October 19, 2016

Nesterenko A., Математические вопросы криптографии 2014 Vol. 5 No. 2 P. 99-102

In this article we present an algorithm for constructing an elliptic curve endomorphism for given complex irrationality. This endomorphism can be used for speeding up a group operation on elliptic curve. ...

Added: February 2, 2015

Buff X., Goncharuk Nataliya, Journal of Modern Dynamics 2015 Vol. 9 P. 169-190

We investigate the notion of complex rotation number which was introduced by V.I.Arnold in 1978. Let f: R/Z -> R/Z be a (real) analytic orientation preserving circle diffeomorphism and let omega in C/Z be a parameter with positive imaginary part. Construct a complex torus by glueing the two boundary components of the annulus { z ...

Added: October 10, 2013

Hutsalyuk A., Liashyk A., Pakuliak S. Z. et al., Russian Mathematical Surveys 2017 Vol. 72 No. 1 P. 33-99

Bethe vectors are found for quantum integrable models associated with the supersymmetric Yangians in terms of the current generators of the Yangian double . The method of projections onto intersections of different types of Borel subalgebras of this infinite-dimensional algebra is used to construct the Bethe vectors. Calculation of these projections makes it possible to express the ...

Added: October 26, 2017

Pavlov A., Mathematische Zeitschrift 2021 No. 297 P. 223-254

We show that for maximal Cohen–Macaulay modules over the homogeneous coordinate ring of a smooth Calabi–Yau varieties X, the computation of Betti numbers can be reduced to computations of dimensions of certain HomHom spaces in the bounded derived category Db(X). In the simplest case of a smooth elliptic curve E embedded in P2 as a smooth cubic, we get explicit values for Betti ...

Added: October 31, 2020

Bogomolov F. A., Fu H., European Journal of Mathematics 2018 Vol. 4 No. 2 P. 555-560

We construct pairs of elliptic curves over number fields with large intersection of projective torsion points. ...

Added: September 13, 2018

Rybnikov L. G., International Mathematics Research Notices 2020 Vol. 2020 No. 22 P. 8766-8785

The Gaudin algebra is the commutative subalgebra in U(g)^⊗N generated by higher integrals of the quantum Gaudin magnet chain attached to a semisimple Lie algebra g. This algebra depends on a collection of pairwise distinct complex numbers z1,…,zN. We prove that this subalgebra has a cyclic vector in the space of singular vectors of the tensor product of ...

Added: November 28, 2020

Slavnov N. A., Zabrodin A., Zotov A., Journal of High Energy Physics 2020 No. 6 P. 123

We obtain a determinant representation of normalized scalar products of onshell and off-shell Bethe vectors in the inhomogeneous 8-vertex model. We consider the case of rational anisotropy parameter and use the generalized algebraic Bethe ansatz approach. Our method is to obtain a system of linear equations for the scalar products, prove its solvability and solve ...

Added: August 24, 2020

Aleksei Ilin, Leonid Rybnikov, Transformation Groups 2021 Vol. 26 No. 2 P. 537-564

The Yangian $Y(\fg)$ of a simple Lie algebra $\fg$ can be regarded as a deformation of two different Hopf algebras: the universal enveloping algebra of the current algebra $U(\fg[t])$ and the coordinate ring of the first congruence subgroup $\mathcal{O}(G_1[[t^{-1}]])$. Both of these algebras are obtained from the Yangian by taking the associated graded with respect ...

Added: April 2, 2021

Leonid Rybnikov, International Mathematics Research Notices 2018 No. 1 P. 202-235

Cactus group is the fundamental group of the real locus of the Deligne–Mumford moduli space of stable rational curves. This group appears naturally as an analog of the braid group in coboundary monoidal categories. We define an action of the cactus group on the set of Bethe vectors of the Gaudin magnet chain corresponding to ...

Added: February 6, 2018

Derbyshev A. E., Povolotsky A. M., Priezzhev V. B., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2015 Vol. 91 P. 022125

The generalized totally asymmetric exclusion process (TASEP) [J. Stat. Mech. (2012) P05014] is an integrable generalization of the TASEP equipped with an interaction, which enhances the clustering of particles. The process interpolates between two extremal cases: the TASEP with parallel update and the process with all particles irreversibly merging into a single cluster moving as ...

Added: February 19, 2015

Serge Lvovski, Moscow Mathematical Journal 2019 Vol. 19 No. 3 P. 597-613

We show that if we are given a smooth non-isotrivial family of curves of genus 1 over C with a smooth base B for which the general fiber of the mapping J : B → A 1 (assigning j-invariant of the fiber to a point) is connected, then the monodromy group of the family (acting ...

Added: August 30, 2019