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Cactus group and monodromy of Bethe vectors
Rybnikov L. G.
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 arbitrary semisimple Lie algebra g. Cactus group appears in our construction as a subgroup in the Galois group of Bethe Ansatz equations. Following the idea of Pavel Etingof, we conjecture that this action is isomorphic to the action of the cactus group on the tensor product of crystals coming from the general coboundary category formalism. We prove this conjecture in the case g=sl2 (in fact, for this case the conjecture almost immediately follows from the results of Varchenko on asymptotic solutions of the KZ equation and crystal bases). We also present some conjectures generalizing this result to Bethe vectors of shift of argument subalgebras and relating the cactus group with the Berenstein-Kirillov group of piecewise-linear symmetries of the Gelfand-Tsetlin polytope.
Publication based on the results of:
Тукачев Н. В., Maslennikov D., Sosorev A. et al., The Journal of Physical Chemistry Letters 2019 Vol. 10 No. 12 P. 3232–3239
Conformational space of polyphenylenevinylene oligomers is systematically investigated computationally at energies relevant for room temperature dynamics in a solvent and in a solid state. Our calculations show that optimal oligomer structures are essentially planar. However, lack of a deep minimum at the planar geometry allows for large molecular deformations even at very low temperatures. At ...
Added: October 14, 2026
Мальцева В. Е., Rozhkov A., Ananyev I. et al., Inorganic Chemistry 2026 Vol. 65 No. 22 P. 12484–12496
Despite the well-established importance of chalcogen bonding in crystal engineering and supramolecular chemistry, experimental charge density data for TeII-centered chalcogen bonds have remained entirely absent, leaving the electronic nature of these interactions uncharacterized at the level of directly measured electron density. Here, we report the first high-resolution X-ray diffraction (HRXRD) study of TeII-centered chalcogen bonds, ...
Added: October 2, 2026
Aleksei Ilin, Kamnitzer J., Leonid Rybnikov, Advances in Mathematics 2025 Vol. 482 No. B Article 110616
We introduce and study the family of trigonometric Gaudin subalgebras in Ug⊗n for arbitrary simple Lie algebra g. This is the family of commutative subalgebras of maximal possible transcendence degree that serve as a universal source for higher integrals of the trigonometric Gaudin quantum spin chain attached to g. We study the parameter space that ...
Added: October 23, 2025
Felder G., Dalipi R., Gurenkova A., Transformation Groups 2024
We give a fermionic formula for $R$-matrices of exterior powers of
the vector representations of $U_q(\widehat{ \mathfrak{gl}}_N)$ and
relate it to the dynamical Weyl group of Tarasov--Varchenko and
Etingof--Varchenko, via a Howe
($\mathfrak{gl}_N,\mathfrak{gl}_M)$-duality. In the limit $N\to\infty$
we obtain $R$-matrices for Fock spaces. As a consequence of our result we
obtain a dynamical action of ...
Added: May 6, 2025
Amburg N., Tolstukhin I., Journal of Geometry and Physics 2025 Vol. 211 Article 105436
We study the three-point quantum sl2 Gaudin model. In this case the compactification of the parameter space is M0,4(C)‾, which is the Riemann sphere. We analyze sphere coverings by the joint spectrum of the Gaudin Hamiltonians treating them as algebraic curves. We write equations for these curves as determinants of tridiagonal matrices and deduce some ...
Added: March 8, 2025
Kalle P., Stanislav I. Bezzubov, Latipov E. et al., Inorganic Chemistry 2025 Vol. 64 No. 4 P. 2146–2153
The crystal structure determines the properties of compounds and materials, although one can find simple yet industrially relevant compounds such as potassium acetate (KOAc) and its hydrates for which the properties and even the composition still remain misunderstood, owing to the lack of structural data. In this study, the crystal structures of KOAc polymorphs and ...
Added: February 25, 2025
Stanislav I. Bezzubov, Yakushev I., Medvedev A. et al., Crystal Growth & Design 2023 Vol. 23 No. 10 P. 7252–7265
A novel class of crystalline hydrogen peroxide adducts, peroxosolvates of zwitterionic sulfonic acids C2H7NO3S·H2O2 (1), p-C6H7NO3S·H2O2 (2), p-C6H7NO3S·0.5(H2O2) (3), m-C6H7NO3S·H2O2 (4), o-C6H7NO3S·H2O2 (5), C5H5NO3S·H2O2 (6), C12H8N2O3S·H2O2 (7), C8H17NO3S·2(H2O2) (8), C10H9NO4S·1.25(H2O2)·0.25(H2O) (9), C10H9NO4S·H2O2 (10), and C10H9NO4S·0.94(H2O2)·0.06(H2O) (11), were prepared from highly concentrated hydrogen peroxide, and their structures were determined by X-ray crystallography. In all compounds, first ...
Added: September 16, 2024
Povolotsky A. M., Journal of Statistical Mechanics: Theory and Experiment 2023 Vol. 2023 No. 3 Article 033103
This work continues the study started in Povolotsky (2021 J. Phys. A: Math. Theor. 54 22LT01), where the exact densities of loops in the O(1) dense loop model on an infinite strip of the square lattice with periodic boundary conditions were obtained. These densities are also equal to the densities of critical percolation clusters on ...
Added: March 25, 2024
B. Feigin, L. Rybnikov, F. Uvarov, Letters in Mathematical Physics 2024 Vol. 114 No. 1 Article 3
We show that the construction of the higher Gaudin Hamiltonians associated with the Lie algebra gl(n) admits an interpolation to any complex number n. We do this using the Deligne’s category D(t), which is a formal way to define the category of finite-dimensional representations of the group GL(n), when n is not necessarily a natural number. We also obtain ...
Added: January 12, 2024
Springer, 2016.
This volume presents modern trends in the area of symmetries and their applications based on contributions from the workshop "Lie Theory and Its Applications in Physics", held near Varna, Bulgaria, in June 2015. Traditionally, Lie theory is a tool to build mathematical models for physical systems.
Recently, the trend has been towards geometrization of the mathematical ...
Added: October 18, 2023
Torubaev Y., Skabitsky I., Konstantin A. Lyssenko, Crystal Growth & Design 2022 Vol. 22 No. 2 P. 1244–1252
Organometallic halogenides Cp2MX2 (M = Ti, Zr; X = Cl, Br, I) were utilized as σ- and π-hole acceptors in interaction with deficient perfluoroarenes. The chain assembly of Cp2MX2 molecules stabilized by H-X hydrogen bonds, characteristic for their native crystals, recur unchanged in their 1:1 cocrystals with 1,4-DITFB. The arrangement between the Cp2MX2 and 1,4-DITFB ...
Added: October 26, 2022
Zabrodin A., Zotov A., Journal of High Energy Physics 2022 No. 7 Article 23
We suggest a field extension of the classical elliptic Ruijsenaars-Schneider model. The model is defined in two different ways which lead to the same result. The first one is via the trace of a chain product of L-matrices which allows one to introduce the Hamiltonian of the model and to show that the model is gauge ...
Added: August 18, 2022
A A Trofimova, A M Povolotsky, Journal of Physics A: Mathematical and Theoretical 2022 Vol. 55 No. 2 Article 025202
We consider the particle current in the asymmetric avalanche process on a ring. It is known to exhibit a transition from the intermittent to continuous flow at the critical density of particles. The exact expressions for the first two scaled cumulants of the particle current are obtained in the large time limit t → ∞ ...
Added: August 15, 2022
А. М. Поволоцкий, Физика элементарных частиц и атомного ядра 2021 Т. 52 № 2 С. 459–529
This review gives a survey of some results about systems of interacting particles and
the laws characterizing their behavior on large scales, which are common for a number of
phenomena uniˇed under the notion of the KardarÄParisiÄZhang universality class. ...
Added: August 15, 2022
Gorsky A., Vasilyev M., Zotov A., Journal of High Energy Physics 2022 Vol. 2022 No. 04 Article 159
In this study we map the dualities observed in the framework of integrable probabilities into the dualities familiar in a realm of integrable many-body systems. The dualities between the pairs of stochastic processes involve one representative from Macdonald-Schur family, while the second representative is from stochastic higher spin six-vertex model of TASEP family. We argue ...
Added: April 28, 2022
Gavrylenko P., Semenyakin M, Zenkevich Y., Journal of High Energy Physics 2021 No. 5 Article 103
We notice a remarkable connection between the Bazhanov-Sergeev solution of Zamolodchikov tetrahedron equation and certain well-known cluster algebra expression. The tetrahedron transformation is then identified with a sequence of four mutations. As an application of the new formalism, we show how to construct an integrable system with the spectral curve with arbitrary symmetric Newton polygon. Finally, we embed this integrable system into the double Bruhat cell of a Poisson-Lie group, show ...
Added: November 26, 2021
Оганов А. Р., Konson G., В кн.: Art History in the Context of Other Sciences in Modern World: Parallels and Interaction.: M.: Information and Publishing House Filin, 2020. С. 278–291.
An interview by Grigoriy Konson with Artem R. Oganov, PhD, FRSC MAE, Professor at the Skolkovo Institute of Science and Technology, Professor at the Russian Academy of Sciences, is devoted to crystallography—the science of solid crystalline substances surrounding us, containing an ordered periodic system of atomic positions. Crystallography is at the forefront of the science ...
Added: May 9, 2021
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