100 research outputs found

### On the Yang-Baxter equation for the six-vertex model

In this paper we review the theory of the Yang-Baxter equation related to the
6-vertex model and its higher spin generalizations. We employ a 3D approach to
the problem. Starting with the 3D R-matrix, we consider a two-layer projection
of the corresponding 3D lattice model. As a result, we obtain a new expression
for the higher spin $R$-matrix associated with the affine quantum algebra
$U_q(\widehat{sl(2)})$. In the simplest case of the spin $s=1/2$ this
$R$-matrix naturally reduces to the $R$-matrix of the 6-vertex model. Taking a
special limit in our construction we also obtain new formulas for the
$Q$-operators acting in the representation space of arbitrary (half-)integer
spin. Remarkably, this construction can be naturally extended to any complex
values of spin $s$. We also give all functional equations satisfied by the
transfer-matrices and $Q$-operators.Comment: 25 pages, 1 figur

### An Analytic Formula for the A_2 Jack Polynomials

In this letter I shall review my joint results with Vadim Kuznetsov and
Evgeny Sklyanin [Indag. Math. 14 (2003), 451-482, math.CA/0306242] on
separation of variables for the $A_n$ Jack polynomials. This approach
originated from the work [RIMS Kokyuroku 919 (1995), 27-34, solv-int/9508002]
where the integral representations for the $A_2$ Jack polynomials was derived.
Using special polynomial bases I shall obtain a more explicit expression for
the $A_2$ Jack polynomials in terms of generalised hypergeometric functions.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA

### Q-operators in the six-vertex model

In this paper we continue the study of $Q$-operators in the six-vertex model
and its higher spin generalizations. In [1] we derived a new expression for the
higher spin $R$-matrix associated with the affine quantum algebra
$U_q(\widehat{sl(2)})$. Taking a special limit in this $R$-matrix we obtained
new formulas for the $Q$-operators acting in the tensor product of
representation spaces with arbitrary complex spin.
Here we use a different strategy and construct $Q$-operators as integral
operators with factorized kernels based on the original Baxter's method used in
the solution of the eight-vertex model. We compare this approach with the
method developed in [1] and find the explicit connection between two
constructions. We also discuss a reduction to the case of finite-dimensional
representations with (half-) integer spins.Comment: 18 pages, no figure

### The eight-vertex model and Painleve VI

In this letter we establish a connection of Picard-type elliptic solutions of
Painleve VI equation with the special solutions of the non-stationary Lame
equation. The latter appeared in the study of the ground state properties of
Baxter's solvable eight-vertex lattice model at a particular point,
$\eta=\pi/3$, of the disordered regime.Comment: 9 pages, LaTeX, submitted to the special issue on Painleve VI,
Journal of Physics

### Elliptic solution for modified tetrahedron equations

As is known, tetrahedron equations lead to the commuting family of
transfer-matrices and provide the integrability of corresponding
three-dimensional lattice models. We present the modified version of these
equations which give the commuting family of more complicated two-layer
transfer-matrices. In the static limit we have succeeded in constructing the
solution of these equations in terms of elliptic functions.Comment: 11 page

### Electromagnetic Structure Functions of Nucleons in the Region of Very Small X

A two component model describing the electromagnetic nucleon structure
functions in the low-x region, based on generalized vector dominance and color
dipole approaches is briefly described.Comment: 3 pages, 1 figure, Talk given at the 14th Lomonosov Conference,
Moscow, August 200

### Eight-vertex model and Painlev\'e VI equation. II. Eigenvector results

We study a special anisotropic XYZ-model on a periodic chain of an odd length
and conjecture exact expressions for certain components of the ground state
eigenvectors. The results are written in terms of tau-functions associated with
Picard's elliptic solutions of the Painlev\'e VI equation. Connections with
other problems related to the eight-vertex model are briefly discussed.Comment: 18 page

### Eight-vertex model and non-stationary Lame equation

We study the ground state eigenvalues of Baxter's Q-operator for the
eight-vertex model in a special case when it describes the off-critical
deformation of the $\Delta=-1/2$ six-vertex model. We show that these
eigenvalues satisfy a non-stationary Schrodinger equation with the
time-dependent potential given by the Weierstrass elliptic P-function where the
modular parameter $\tau$ plays the role of (imaginary) time. In the scaling
limit the equation transforms into a ``non-stationary Mathieu equation'' for
the vacuum eigenvalues of the Q-operators in the finite-volume massive
sine-Gordon model at the super-symmetric point, which is closely related to the
theory of dilute polymers on a cylinder and the Painleve III equation.Comment: 11 pages, LaTeX, minor misprints corrected, references adde

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