arXiv subjects
Publications and source records attributed to S. Pakuliak.
The factorization of the universal R-matrix corresponding to so called Drinfeld Hopf structure is described on the example of quantum affine algebra $U_q(\hat{sl}_2)$. As a result of factorization procedure we deduce certain differential equations on the factors of the universal ${\cal R}$-matrix, which allow to construct uniquely these factors in the integral form.
The goal of this paper is to analyse the method of angular quantization for the Sine-Gordon model at the free fermion point, which is one of the most investigated models of the two-dimensional integrable field theories. The angular quantization method (see hep-th/9707091) is a continuous analog of the Baxter's corner transfer matrix method. Investigating the canonical quantization of the free massive Dirac fermions in one Rindler wedge we identify this quantization with a representation of the infinite-dimensional algebra introduced in the paper q-alg/9702002 and specialized to the free fermion point. We construct further the main ingredients of the SG theory in terms of the representation theory of this algebra following the approach by M.Jimbo, T.Miwa et al.
The classical limit of the scaled elliptic algebra $A_{\hbar,η}(sl_2)$ is investigated. The limiting Lie algebra is described in two equivalent ways: as a central extension of the algebra of generalized automorphic $sl_2$ valued functions on a strip and as an extended algebra of decreasing automorphic $sl_2$ valued functions on the real line. A bialgebra structure and an infinite-dimensional representation in the Fock space are studied. The classical limit of elliptic algebra $A_{q,p}(sl_2)$ is also briefly presented.
We investigate different Hopf algebras associated to Yang's solution of quantum Yang-Baxter equation. It is shown that for the precise definition of the algebra one needs the commutation relations for the deformed algebra of formal currents and the specialization of the Riemann problem for the currents. Two different Riemann problems are considered. They lead to the central extended Yangian double associated with ${sl}_2$ and to the degeneration of scaling limit of elliptic affine algebra. Unless the defining relations for the generating functions of the both algebras coincide their properties and the theory of infinite-dimensional representations are quite different. We discuss also the Riemann problem for twisted algebras and for scaled elliptic algebra.
We discuss relations between different formulae for solutions of the Knizhnik-Zamolodchikov differential and the quantum Knizhnik-Zamolodchikov difference equations at level 0 and associated with rational solutions of the Yang-Baxter equation.
The scaling limit $A_{\hbar,η}(\hat{sl_2})$ of the elliptic algebra $A_{q,p}(\hat{sl_2})$ is investigated. The limiting algebra is defined in terms of a continuous family of generators being Fourier harmonics of Gauss coordinates of the $L$-operator. Ding-Frenkel isomorphism between $L$-operator's and current descriptions of the algebra $A_{\hbar,η}(\hat{sl_2})$ is established and is identified with the Riemann problem on a strip. The representations, coalgebraic structure and intertwining operators of the algebra are studied.
We continue the investigation of the central extended Yangian double [S. Khoroshkin, q-alg/9602031]. In this paper we study the intertwining operators for certain infinite dimensional representations of $\Yd$, which are deformed analogs of the highest weight representations of the affine algebra $\aff$ at level 1. We give bosonized expressions for intertwining operators, verify that they generate an algebra isomorphic to Zamolodchikov--Faddeev algebra for the $SU(2)$-invariant Thirring model. We compose from them $L$-operators by Miki's prescription and verify that they coincide with $L$-operators constructed from universal ${\cal R}$-matrix.
The traces over infinite dimensional representations of the central extended Yangian double for the product of operators which intertwine these representations are calculated. For the special combinations of the intertwining operators the traces are identified with form factors of local operators in SU(2)-invariant Thirring model. This identification is based on the identities which are deformed analogs of the Gauss-Manin connection identities for the hyperelliptic curves.
A generalized AKNS systems introduced and discussed recently in \cite{dGHM} are considered. It was shown that the dressing technique both in matrix pseudo-differential operators and formal series with respect to the spectral parameter can be developed for these hierarchies.
Some relations between different objects associated with quantum affine algebras are reviewed. It is shown that the Frenkel-Jing bosonization of a new realization of quantum affine algebra $\Uqa$ as well as bosonization of $L$-operators for this algebra can be obtained from Zamolodchikov-Faddeev algebras defined by the quantum $R$-matrix satisfying unitarity and crossing-symmetry conditions. (Talk given at the International Coference "Modern Problems of Quantum Field Theory, Quantum Gravity and Strings", Alushta, June 10--20, 1994.)
A simple property of the integrals over the hyperelliptic surfaces of arbitrary genus is observed. Namely, the derivatives of these integrals with respect to the branching points are given by the linear combination of the same integrals. We check that this property is responsible for the solution to the level zero Knizhnik-Zamolodchikov equation given in terms of hyperelliptic integrals.
The annihilation poles for the form factors in XXZ model are studied using vertex operators introduced in \cite{DFJMN}. An annihilation pole is the property of form factors according to which the residue of the $2n$-particle form factor in such a pole can be expressed through linear combination of the $2n-2$-particle form factors. To prove this property we use the bosonization of the vertex operators in XXZ model which was invented in \cite{JMMN}.
The double scaling limit of a new class of the multi-matrix models proposed in \cite{MMM91}, which possess the $W$-symmetry at the discrete level, is investigated in details. These models are demonstrated to fall into the same universality class as the standard multi-matrix models. In particular, the transformation of the W-algebra at the discrete level into the continuum one of the paper \cite{FKN91a} is proposed, the corresponding partition functions being compared. All calculations are demonstrated in full in the first non-trivial case of $W^{(3)}$-constraints.
We introduce {\it conformal multi-matrix models} (CMM) as an alternative to conventional multi-matrix model description of two-dimensional gravity interacting with $c < 1$ matter. We define CMM as solutions to (discrete) extended Virasoro constraints. We argue that the so defined alternatives of multi-matrix models represent the same universality classes in continuum limit, while at the discrete level they provide explicit solutions to the multi-component KP hierarchy and by definition satisfy the discrete $W$-constraints. We prove that discrete CMM coincide with the $(p,q)$-series of 2d gravity models in a {\it well}-{\it defined} continuum limit, thus demonstrating that they provide a proper generalization of Hermitian one-matrix model.