arXiv subjects

R. M. Kashaev

Publications and source records attributed to R. M. Kashaev.

26 records · Page 2Linked to original sources

arXiv1995

On Discrete 3-Dimensional Equations Associated with the Local Yang-Baxter Relation

The local Yang-Baxter equation (YBE), introduced by Maillet and Nijhoff, is a proper generalization to 3 dimensions of the zero curvature relation. Recently, Korepanov has constructed an infinite set of integrable 3-dimensional lattice models, and has related them to solutions to the local YBE. The simplest Korepanov's model is related to the star-triangle relation in the Ising model. In this paper the corresponding discrete equation is derived. In the continuous limit it leads to a differential 3d equation, which is symmetric with respect to all permutations of the three coordinates. A similar analysis of the star-triangle transformation in electric networks leads to the discrete bilinear equation of Miwa, associated with the BKP hierarchy. Some related operator solutions to the tetrahedron equation are also constructed.

solv-int↗

arXiv1995

Affine Toda field theory as a 3-dimensional integrable system

The affine Toda field theory is studied as a 2+1-dimensional system. The third dimension appears as the discrete space dimension, corresponding to the simple roots in the $A_N$ affine root system, enumerated according to the cyclic order on the $A_N$ affine Dynkin diagram. We show that there exists a natural discretization of the affine Toda theory, where the equations of motion are invariant with respect to permutations of all discrete coordinates. The discrete evolution operator is constructed explicitly. The thermodynamic Bethe ansatz of the affine Toda system is studied in the limit $L,N\to\infty$. Some conjectures about the structure of the spectrum of the corresponding discrete models are stated.

hep-th↗

arXiv1995

A Link Invariant from Quantum Dilogarithm

The link invariant, arising from the cyclic quantum dilogarithm via the particular $R$-matrix construction, is proved to coincide with the invariant of triangulated links in $S^3$ introduced in R.M. Kashaev, Mod. Phys. Lett. A, Vol.9 No.40 (1994) 3757. The obtained invariant, like Alexander-Conway polynomial, vanishes on disjoint union of links. The $R$-matrix can be considered as the cyclic analog of the universal $R$-matrix associated with $U_q(sl(2))$ algebra.

q-alg↗

arXiv1995

Heisenberg Double and Pentagon Relation

It is shown that the Heisenberg double has a canonical element, satisfying the pentagon relation. From a given invertible constant solution to the pentagon relation one can restore the structure of the underlying algebras. Drinfeld double can be realized as a subalgebra in the tensor square of the Heisenberg double. This enables one to write down solutions to the Yang-Baxter relation in terms of solutions to the pentagon relation.

q-alg↗

arXiv1994

Quantum Dilogarithm as a 6j-Symbol

The cyclic quantum dilogarithm is interpreted as a cyclic 6j-symbol of the Weyl algebra, considered as a Borel subalgebra $BU_q(sl(2))$. Using modified 6j-symbols, an invariant of triangulated links in triangulated 3-manifolds is constructed. Apparently, it is an ambient isotopy invariant of links.

hep-th↗

arXiv1993

Generalized Bethe Ansatz Equations for Hofstadter Problem

The problem of diagonalization of the quantum mechanical Hamiltonian, governing dynamics of an electron on a two-dimensional triangular or square lattice in external uniform magnetic field, applied perpendicularly to the lattice plane, the flux through lattice cell, divided by the elementary quantum flux, being rational number, is reduced to generalized Bethe ansatz like equations on high genus algebraic curve. Our formulae for the trigonometric case, where genus of the curve vanishes, contain as a particular case recent result of Wiegmann and Zabrodin.

hep-th↗

arXiv1993

Quantum Dilogarithm

A quantum generalization of Rogers' five term, or ``pentagon'' dilogarithm identity is suggested. It is shown that the classical limit gives usual Rogers' identity. The case where the quantum identity is realized in finite dimensional space is also considered and the quantum dilogarithm is constructed as a function on Fermat curve, while the identity itself is equivalent to the restricted star-triangle relation introduced by Bazhanov and Baxter.

hep-th↗

arXiv1993

Generalized Yang-Baxter Equation

A generalization of the Yang-Baxter equation is proposed. It enables to construct integrable two-dimensional lattice models with commuting two-layer transfer matrices, while single-layer ones are not necessarily commutative. Explicit solutions to the generalized equations are found. They are related with Botzmann weights of the $sl(3)$ chiral Potts model.

hep-th↗