A matrix solution to pentagon equation with anticommuting variables

We construct a solution to pentagon equation with anticommuting variables living on two-dimensional faces of tetrahedra. In this solution, matrix coordinates are ascribed to tetrahedron vertices. As matrix multiplication is noncommutative, this provides a "more quantum" topological field theory than in our previous works

Quantum 2+1 evolution model

A quantum evolution model in 2+1 discrete space - time, connected with 3D fundamental map R, is investigated. Map R is derived as a map providing a zero curvature of a two dimensional lattice system called "the current system". In a special case of the local Weyl algebra for dynamical variables the map appears to be canonical one and it corresponds to known operator-valued R-matrix. The current system is a kind of the linear problem for 2+1 evolution model. A generating function for the integrals of motion for the evolution is derived with a help of the current system. The subject of the paper is rather new, and so the perspectives of further investigations are widely discussed.Comment: LaTeX, 37page

Ground states of Heisenberg evolution operator in discrete three-dimensional space-time and quantum discrete BKP equations

In this paper we consider three-dimensional quantum q-oscillator field theory without spectral parameters. We construct an essentially big set of eigenstates of evolution with unity eigenvalue of discrete time evolution operator. All these eigenstates belong to a subspace of total Hilbert space where an action of evolution operator can be identified with quantized discrete BKP equations (synonym Miwa equations). The key ingredients of our construction are specific eigenstates of a single three-dimensional R-matrix. These eigenstates are boundary states for hidden three-dimensional structures of U_q(B_n^1) and U_q(D_n^1)$.Comment: 13 page

Upgrade of the Argentine Islands INTERMAGNET observatory at Akademik Vernadsky station, Antarctica

The article describes the main features of upgrading the magnetometric complex based on the LEMI-025 variometer in January-April 2019 at the geomagnetic observatory (code AIA) of the Ukrainian Antarctic Akademik Vernadsky station.У статті описані основні особливості модернізації магнітометричного комплексу на базі варіометра LEMI-025 у січні-квітні 2019 року в геомагнітній обсерваторії (код AIA) Української антарктичної станції «Академік Вернадський»

Permutation-type solutions to the Yang-Baxter and other n-simplex equations

We study permutation type solutions to n-simplex equations, that is, solutions whose R matrix can be written as a product of delta- functions depending linearly on the indices. With this ansatz the D^{n(n+1)} equations of the n-simplex equation reduce to an [n(n+1)/2+1]x[n(n+1)/2+1] matrix equation over Z_D. We have completely analyzed the 2-, 3- and 4-simplex equations in the generic D case. The solutions show interesting patterns that seem to continue to still higher simplex equations.Comment: 20 pages, LaTeX2e. to appear in J. Phys. A: Math. Gen. (1997

Tetrahedron and 3D reflection equations from quantized algebra of functions

Soibelman's theory of quantized function algebra A_q(SL_n) provides a representation theoretical scheme to construct a solution of the Zamolodchikov tetrahedron equation. We extend this idea originally due to Kapranov and Voevodsky to A_q(Sp_{2n}) and obtain the intertwiner K corresponding to the quartic Coxeter relation. Together with the previously known 3-dimensional (3D) R matrix, the K yields the first ever solution to the 3D analogue of the reflection equation proposed by Isaev and Kulish. It is shown that matrix elements of R and K are polynomials in q and that there are combinatorial and birational counterparts for R and K. The combinatorial ones arise either at q=0 or by tropicalization of the birational ones. A conjectural description for the type B and F_4 cases is also given.Comment: 26 pages. Minor correction

Finite-dimensional analogs of string s <-> t duality and pentagon equation

We put forward one of the forms of functional pentagon equation (FPE), known from the theory of integrable models, as an algebraic explanation to the phenomenon known in physics as st duality. We present two simple geometrical examples of FPE solutions, one of them yielding in a particular case the well-known Veneziano expression for 4-particle amplitude. Finally, we interpret our solutions of FPE in terms of relations in Lie groups.Comment: LaTeX, 12 pages, 6 eps figure

Dual Resonance Model Solves the Yang-Baxter Equation

The duality of dual resonance models is shown to imply that the four point string correlation function solves the Yang-Baxter equation. A reduction of transfer matrices to $A_l$ symmetry is described by a restriction of the KP $\tau$ function to Toda molecules.Comment: 10 pages, LaTe

The Q-operator for Root-of-Unity Symmetry in Six Vertex Model

We construct the explicit $Q$-operator incorporated with the $sl_2$-loop-algebra symmetry of the six-vertex model at roots of unity. The functional relations involving the $Q$-operator, the six-vertex transfer matrix and fusion matrices are derived from the Bethe equation, parallel to the Onsager-algebra-symmetry discussion in the superintegrable $N$-state chiral Potts model. We show that the whole set of functional equations is valid for the $Q$-operator. Direct calculations in certain cases are also given here for clearer illustration about the nature of the $Q$-operator in the symmetry study of root-of-unity six-vertex model from the functional-relation aspect.Comment: Latex 26 Pages; Typos and small errors corrected, Some explanations added for clearer presentation, References updated-Journal version with modified labelling of sections and formula

Two-State Spectral-Free Solutions of Frenkel-Moore Simplex Equation

Whilst many solutions have been found for the Quantum Yang-Baxter Equation (QYBE), there are fewer known solutions available for its higher dimensional generalizations: Zamolodchikov's tetrahedron equation (ZTE) and Frenkel and Moore's simplex equation (FME). In this paper, we present families of solutions to FME which may help us to understand more about higher dimensional generalization of QYBE.Comment: LaTeX file. Require macros: cite.sty and subeqnarray.sty to process. To appear in J. Phys. A: Math. and Ge