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Solution of tetrahedron equation and cluster algebras
Journal of High Energy Physics. 2021. No. 5. Article 103.
Gavrylenko P., Semenyakin M, Zenkevich Y.
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 how triangular decomposition can be used to extend our approach to the general non-symmetric Newton polygons, and prove the Lemma which classifies conjugacy classes in double affine Weyl groups of A-type by decorated Newton polygons.
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
Tsuboi Z., Zabrodin A., Zotov A., Journal of High Energy Physics 2015 Vol. 2015 No. 5, Article number 86
For integrable inhomogeneous supersymmetric spin chains (generalized graded magnets) constructed employing Y(gl(N|M))-invariant R-matrices in finite-dimensional representations we introduce the master T-operator which is a sort of generating function for the family of commuting quantum transfer matrices. Any eigenvalue of the master T-operator is the tau-function of the classical mKP hierarchy. It is a polynomial in ...
Added: September 7, 2015
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
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
Zabrodin A., Alexandrov A., Leurent S. et al., Nuclear Physics B 2014 Vol. 883 No. P. 173-223
Following the approach of [Alexandrov A., Kazakov V., Leurent S., Tsuboi Z., Zabrodin A., J. High Energy Phys. 2013 (2013), no. 9, 064, 65 pages, arXiv:1112.3310], we construct the master T-operator for the quantum Gaudin model with twisted boundary conditions and show that it satisfies the bilinear identity and Hirota equations for the classical KP ...
Added: July 15, 2014
Feigin B. L., Awata H., Shiraishi J., Journal of High Energy Physics 2012 No. 3 P. 41-68
We establish the equivalence between the refined topological vertex of Iqbal-Kozcaz-Vafa and a certain representation theory of the quantum algebra of type W1+∞ introduced by Miki. Our construction involves trivalent intertwining operators Φ and Φ* associated with triples of the bosonic Fock modules. Resembling the topological vertex, a triple of vectors ∈ Z2 is attached ...
Added: September 20, 2012
Zabrodin A., Alexandrov A., Journal of Geometry and Physics 2013 Vol. 67 P. 37-80
We review the formalism of free fermions used for construction of tau-functions of classical integrable hierarchies and give a detailed derivation of group-like properties of the normally ordered exponents, transformations between different normal orderings, the bilinear relations, the generalized Wick theorem and the bosonization rules. We also consider various examples of tau-functions and give their ...
Added: February 16, 2013
Gavrylenko P., Marshakov A., Journal of High Energy Physics 2014 No. 5 P. 97
We study the extended prepotentials for the S-duality class of quiver gauge theories, considering them as quasiclassical tau-functions, depending on gauge theory condensates and bare couplings. The residue formulas for the third derivatives of extended prepotentials are proven, which lead to effective way of their computation, as expansion in the weak-coupling regime. We discuss also ...
Added: October 20, 2014
Mironov A., Morozov A., Natanzon S. M., Journal of High Energy Physics 2011 No. 11(097) P. 1-31
Correlators in topological theories are given by the values of a linear form on the products of operators from a commutative associative algebra (CAA). As a corollary, partition functions of topological theory always satisfy the generalized WDVV equations of. We consider the Hurwitz partition functions, associated in this way with the CAA of cut-and-join operators. ...
Added: October 12, 2012
Derkachov S. E., Olivucci E., Journal of High Energy Physics 2021 Article 146
In this paper we study a wide class of planar single-trace four point correlators in the chiral conformal field theory (χCFT4) arising as a double scaling limit of the γ-deformed NN = 4 SYM theory. In the planar (t’Hooft) limit, each of such correlators is described by a single Feynman integral having the bulk topology of a square ...
Added: November 25, 2021
Rybnikov L. G., / Cornell University. Series QA "arXiv math". 2016. No. 1409.0131.
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: September 29, 2014
Sergey Derkachov, Ferrando G., Olivucci E., Journal of High Energy Physics 2021 Vol. 2021 No. 12 Article 174
We present a basis of eigenvectors for the graph building operators acting along the mirror channel of planar fishnet Feynman integrals in d-dimensions. The eigenvectors of a fishnet lattice of length N depend on a set of N quantum numbers (uk , lk ), each associated with the rapidity and bound-state index of a lattice excitation. Each excitation is a particle in ...
Added: October 27, 2022
Pyatov P. N., Journal of Physics A: Mathematical and Theoretical 2016 Vol. 49 No. 41 P. 415202-(25pp)
We develop a construction of the unitary type anti-involution for the quantized differential calculus over GLq (n) in the case ∣q∣ = 1. To this end, we consider a joint associative algebra of quantized functions, differential forms and Lie derivatives over GLq (n)/SLq (n), which is bicovariant with respect to GLq (n)/SLq (n) coactions. We ...
Added: September 30, 2016
Pogrebkov A., Mathematics 2021 Vol. 9 No. 16 Article 1988
The Kadomtsev–Petviashvili equation is known to be the leading term of a semi-infinite hierarchy of integrable equations with evolutions given by times with positive numbers. Here, we introduce new hierarchy directed to negative numbers of times. The derivation of such systems, as well as the corresponding hierarchy, is based on the commutator identities. This approach ...
Added: October 8, 2021
Marshakov A., Journal of High Energy Physics 2013 Vol. 07 P. 086
The prepotentials for the quiver supersymmetric gauge theories are defined as quasiclassical tau-functions, depending on two different sets of variables: the parameters of the UV gauge theory or the bare compexified couplings, and the vacuum condensates of the theory in IR. The bare couplings are introduced as periods on the UV base curve, and the ...
Added: March 11, 2015
A. Levin, Olshanetsky M., Zotov A., Journal of High Energy Physics 2014 Vol. 2014 No. 10:109 P. 1-29
We construct special rational gl N Knizhnik-Zamolodchikov-Bernard (KZB) equations with Ñ punctures by deformation of the corresponding quantum gl N rational R-matrix. They have two parameters. The limit of the first one brings the model to the ordinary rational KZ equation. Another one is τ. At the level of classical mechanics the deformation parameter τ ...
Added: January 22, 2015
Nirov Khazret S., Razumov A., Journal of Physics A: Mathematical and Theoretical 2017 Vol. 50 No. 305201 P. 1-19
The Verma modules over the quantum groups Uq(gll+1) for arbitrary values
of l are analysed. The explicit expressions for the action of the generators
on the elements of the natural basis are obtained. The corresponding
representations of the quantum loop algebras Uq(L(sll+1)) are constructed
via Jimbo’s homomorphism. This allows us to find certain representations of
the positive Borel subalgebras of ...
Added: January 29, 2018
Alexandrov A., Mironov A., Morozov A. et al., Journal of High Energy Physics 2014 Vol. 11 No. 80 P. 1-31
There is now a renewed interest to a Hurwitz tau-function, counting the
isomorphism classes of Belyi pairs, arising in the study of equilateral triangulations and
Grothiendicks’s dessins d’enfant. It is distinguished by belonging to a particular family
of Hurwitz tau-functions, possessing conventional Toda/KP integrability properties. We
explain how the variety of recent observations about this function fits into the ...
Added: December 2, 2014
Ogievetsky O., Pyatov P. N., Journal of Geometry and Physics 2021 Vol. 162 Article 104086
A notion of quantum matrix (QM-) algebra generalizes and unifies two famous families of algebras from the theory of quantum groups: the RTT-algebras and the reflection equation (RE-) algebras. These algebras being generated by the components of a `quantum' matrix $M$ possess certain properties which resemble structure theorems of the ordinary matrix theory. It turns ...
Added: December 27, 2020
Grekov A., Sechin I., Zotov A., Journal of High Energy Physics 2019 Vol. 10 No. 081 P. 1-32
We introduce a family of classical integrable systems describing dynamics of M interacting glN integrable tops. It extends the previously known model of interacting elliptic tops. Our construction is based on the GLNR-matrix satisfying the associative Yang-Baxter equation. The obtained systems can be considered as extensions of the spin type Calogero-Moser models with (the classical ...
Added: October 15, 2019
Gavrylenko P., Journal of High Energy Physics 2015 No. 09 P. 167
We study the solution of the Schlesinger system for the 4-point $\mathfrak{sl}_N$ isomonodromy problem and conjecture an expression for the isomonodromic τ-function in terms of 2d conformal field theory beyond the known N = 2 Painlevé VI case. We show that this relation can be used as an alternative definition of conformal blocks for the ...
Added: October 9, 2015
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
Zabrodin A., Proceedings of Physics and Mathematics of Nonlinear Phenomena 2014 Vol. 482 No. 012047 P. 10
This short note is a review of the intriguing connection between the quantum Gaudin model and the classical KP hierarchy recently established in [A.Alexandrov, S.Leurent, Z.Tsuboi, A.Zabrodin, The master T-operator for the Gaudin model and KP hierarchy, Nuclear Physics B 883 (2014) 173-223]. We construct the generating function of integrals of motion for the quantum ...
Added: July 15, 2014
Feigin B. L., Feigin E., Jimbo M. et al., Letters in Mathematical Physics 2009 Vol. 88 No. 1-3 P. 39-77
We use the Whittaker vectors and the Drinfeld Casimir element to show that eigenfunctions of the difference Toda Hamiltonian can be expressed via fermionic formulas. Motivated by the combinatorics of the fermionic formulas we use the representation theory of the quantum groups to prove a number of identities for the coefficients of the eigenfunctions. ...
Added: January 25, 2013