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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.
Felder G., Dalipi R., 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
Zabrodin A., Mathematical Physics Analysis and Geometry 2024 Vol. 27 Article 1
We consider multi-component Kadomtsev-Petviashvili hierarchy of type C (the multi-component CKP hierarchy) originally defined with the help of matrix pseudo-differential operators via the Lax-Sato formalism. Starting from the bilinear relation for the wave functions, we prove existence of the tau-function for the multi-component CKP hierarchy and provide a formula which expresses the wave functions through ...
Added: November 28, 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
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
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
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., Marshakov A., Stoyan A., Journal of High Energy Physics 2020 Vol. 12 Article 125
We study the relation of irregular conformal blocks with the Painlevé III3 equation. The functional representation for the quasiclassical irregular block is shown to be consistent with the BPZ equations of conformal field theory and the Hamilton-Jacobi approach to Painlevé III3. It leads immediately to a limiting case of the blow-up equations for dual Nekrasov partition ...
Added: March 24, 2022
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
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
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
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
Genz V., Koshevoy Gleb, Schumann B., Advances in Mathematics 2020 Vol. 369 P. 107178
We establish the relation of Berenstein–Kazhdan’s decoration function and Gross–Hacking–Keel–Kontsevich’s potential on the open double Bruhat cell in the base affine space G/N of a simple, simply connected, simply laced algebraic group G. As a byproduct we derive explicit identifications of polyhedral parametrization of canonical bases of the ring of regular functions on G/N arising ...
Added: May 19, 2020
Gavrylenko P., Santachiara R., Journal of High Energy Physics 2019 Vol. 2019 No. 11 P. 1–36
We present an approach that gives rigorous construction of a class of crossing invariant functions in c = 1 CFTs from the weakly invariant distributions on the moduli space \( {\mathcal{M}}_{0,4}^{\mathrm{SL}\left(s,\mathbb{C}\right)} \) of SL(2, ℂ) flat connections on the sphere with four punctures. By using this approach we show how to obtain correlation functions in the Ashkin-Teller and the Runkel- ...
Added: May 14, 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
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
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
Gavrylenko P., Marshakov A., Journal of High Energy Physics 2016 Vol. 2016 No. 2 P. 1–31
We consider the conformal blocks in the theories with extended conformal W-symmetry for the integer Virasoro central charges. We show that these blocks for the generalized twist fields on sphere can be computed exactly in terms of the free field theory on the covering Riemann surface, even for a non-abelian monodromy group. The generalized twist ...
Added: June 14, 2016
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
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