Three-dimensional integrable models and associated tangle invariants

In this paper we show that the Boltzmann weights of the three-dimensional Baxter-Bazhanov model give representations of the braid group if some suitable spectral limits are taken. In the trigonometric case we classify all possible spectral limits which produce braid group representations. Furthermore, we prove that for some of them we get cyclotomic invariants of links and for others we obtain tangle invariants generalizing the cyclotomic ones.     

Gaudin Model,KZ Equation and an Isomonodromic Problem on the Tours

This Letter presents a construction of isospectral problems on the torus. The construction starts from an SU(n) version of the XYZ Gaudin model recently studied by Kuroki and Takebe within the context of a twisted WZW model. In the classical limit, the quantum Hamiltonians of the generalized Gaudin model turn into classical Hamiltonians with a natural r-matrix structure. These Hamiltonians are used to build a nonautonomous multi-time Hamiltonian system, which is eventually shown to be an isomonodromic problem on the torus. This isomonodromic problem can also be reproduced from an elliptic analogue of the KZ equation for the twisted WZW model. Finally, a geometric interpretation of this isomonodromic problem is discussed in the language of a moduli space of meromorphic connections.     

Selberg-Type Integrals Associated with SL3

We present several formulae for Selberg-type integrals associated with the Lie algebra .     

Generalization of the Knizhnik–Zamolodchikov Equations

In this Letter, we introduce a generalization of the Knizhnik–Zamolodchikov equations from affine Lie algebras to a wide class of conformal field theories (not necessarily rational). The new equations describe correlations functions of primary fields and of a finite number of their descendents. Our proposal is based on Nahm's concept of small spaces which provide adequate substitutes for the lowest energy subspaces in modules of affine Lie algebras. We explain how to construct the first order differential equations and investigate properties of the associated connections, thereby preparing the grounds for an analysis of quantum symmetries. The general considerations are illustrated in examples of Virasoro minimal models.     

Duality for Knizhnik–Zamolodchikov and Dynamical Equations

We consider the Knizhnik–Zamolodchikov (KZ) and dynamical equations, both differential and difference, in the context of (gl _k ,gl _n ) duality. We show that the KZ and dynamical equations naturally exchange under the duality.     

New solvable lattice models in three dimensions

In this paper we establish a remarkable connection between two seemingly unrelated topics in the area of solvable lattice models. The first is the Zamolodchikov model, which is the only nontrivial model on a three-dimen-sional lattice so far solved. The second is the chiral Potts model on the square lattice and its generalization associated with theU _q(sl(n)) algebra, which is of current interest due to its connections with high-genus algebraic curves and with representations of quantum groups at roots of unity. We show that this last sl(n)-generalized chiral Potts model can be interpreted as a model on a threedimensional simple cubic lattice consisting ofn square-lattice layers with anN- valued (N2) spin at each site. Further, in theN=2 case this three-dimen-sional model reduces (after a modification of the boundary conditions) to the Zamolodchikov model we mentioned above.     

Dynamical Differential Equations Compatible with Rational <Emphasis Type="Italic">qKZ</Emphasis> Equations

For the Lie algebra _N we introduce a system of differential operators called the dynamical operators. We prove that the dynamical differential operators commute with the _N rational quantized Knizhnik–Zamolodchikov difference operators. We describe the transformations of the dynamical operators under the natural action of the _N Weyl group.Mathematics Subject Classifications (2000). 17B37, 17B80, 81R10.     

Star-triangle relation for a three-dimensional model

The solvablesl(n)-chiral Potts model can be interpreted as a three-dimensional lattice model with local interactions. To within a minor modification of the boundary conditions it is an Ising-type model on the body-centered cubic lattice with two- and three-spin interactions. The corresponding local Boltzmann weights obey a number of simple relations, including a restricted star-triangle relation, which is a modified version of the well-known star-triangle relation appearing in two-dimensional models. We show that these relations lead to remarkable symmetry properties of the Boltzmann weight function of an elementary cube of the lattice, related to the spatial symmetry group of the cubic lattice. These symmetry properties allow one to prove the commutativity of the row-to-row transfer matrices, bypassing the tetrahedron relation. The partition function per site for the infinite lattice is calculated exactly.On leave of absence from the Institute for High Energy Physics, Protvino, Moscow Region, 142284, Russia.     

The vertex formulation of the Bazhanov-Baxter model

In this paper we formulate an integrable model on the simple cubic lattice. TheN-valued spin variables of the model belong to edges of the lattice. The Boltzmann weights of the model obey the vertex-type tetrahedron equation. In the thermodynamic limit our model is equivalent to the Bazhanov-Baxter model. In the case whenN=2 we reproduce Korepanov's and Hietarinta's solutions of the tetrahedron equation as special cases.     

Regular Basis and R-Matrices for the ^(su)(n)k Knizhnik–Zamolodchikov Equation

Dynamical R-matrix relations are derived for the group-valued chiral vertex operators in the SU(n) WZNW model from the KZ equation for a general four-point function including two step operators. They fit the exchange relations of the U _q (sl _n ) covariant quantum matrix algebra derived previously by solving the dynamical Yang–Baxter equation. As a byproduct, we extend the regular basis introduced earlier for SU(2) chiral fields to SU(n) step operators and display the corresponding triangular matrix representation of the braid group.