International Journal of Modern Physics A

All eigenvalues of the transfer matrix of the integrable 3-state Potts model are computed as polynomials in the spectral variable for chains of length M≤7. The zeroes of the eigenvalues are known to satisfy a Bethe's Ansatz equation and thus it is of particular interest that we find many solutions whose zeroes do not satisfy the traditional "string hypothesis". We also find many cases where the integers in the logarithmic form of the Bethe equations do not satisfy the monotonicity properties that they are usually assumed to possess. We present a classification of all eigenvalues in terms of sets of roots and show that, for all M, this classification yields a complete set.

We show that the symmetry algebra of the SL(2,R)_k/ U(1) coset model is a non-linear deformation of W_∞, characterized by k. This is a universal W-algebra which linearizes in the large k limit and truncates to W_N for K=-N. Using the theory of non-compact parafermions we construct a free field realization of the non-linear W_∞ in terms of two bosons with background charge. The W-characters of all unitary SL(2,R)/ U(1) representations are computed. Applications to the physics of 2-d black hole backgrounds are also discussed and connections with the KP approach to c=1 string theory are outlined.

We compute the short distance asymptotics of a tau-fucntion appearing in the work of Sato, Miwa, and Jimbo on holonomic quantum fields, We show that these asymptotics are determined by the Widom operator. This same operator is fundamental in the asymptotics of Toeplitz determinants with singular generating fucntions.

We demonstrate that the quantization of momenta in different two dimensional theories of elementary particles with factorizable scattering amplitudes gives the so-called degenerate affine Hecke algebras and their versions. Some connections with quantum groups(Yangians), the two-dimensional conformal field theory and representation theory am discussed. In particular, an interpretation and generalizations of the difference counterpart of the Knizhnik-Zamolodchikov equation are found by means of the particles on a segment.

This paper is a continuation of the papers [DC-K] and [DC-K-P] on representations of quautum groups at roots of 1. Here we show that an irreducible representation of a quantum group at an odd root of 1 can be uniquely induced from an exceptional representation of a smaller quantum group. This reduces the classification of representations, the calculation of their characters and dimensions, etc, to the exceptional case.

We compute the mean staggered polarization of the vertex models corresponding to higher spin representations of . This generalizes a result of Baxter for spin 1/2.

We analyse the transfer matrix spectra of vertex models associated with the Lie superalgebras gl(P|M) and sl(P|M) using the representation theory of the Hecke algebra H_n(q). We develop a terminology for discussing the Bethe ansatz computation of the spectrum from this perspective. Using representations coming from these vertex models we develop some new methods for dealing with the analysis of Hecke algebras in any specialisation, including roots of unity. We also discuss the construction and spectrum of fusion models from the viewpoint of representation theory, begining a classification of the spectrum and identifying some sectors with trivial spectrum. In particular we show that the spectrum of the sl(P|M) λ-fusion model is trivial if λ_P+1>M.

We prove Drinfeld's conjecture that the center of a certain completion of the universal enveloping algebra of an affine Kac-Moody algebra at the critical level is isomorphic to the Gelfand-Dikii algebra, associated to the Langlands dual algebra. The center is identified with a limit of the W-algebra, defined by means of the quantum Drinfeld-Sokolov reduction.

We describe several aspects of the annihilating ideals and reduced chiral algebras of conformal field theories, especially, minimal models of W_n algebras. The structure of the annihilating ideal and a vanishing condition is given. Using the annihilating ideal, the structure of quasi-finite models of the Virasoro (2,q) minimal models are studied, and their intimate relation to the Gordon identities are discussed. We also show the examples in which the reduced algebras of W_n and W_ℓ algebras at the same central charge are isomorphic to each other.

We study the structure of superselection sectors of an arbitrary perturbation of a conformal field theory. We describe how a restriction of the q-deformed sl(2) affine Lie algebra symmetry of the sine-Gordon theory can be used to derive the S-matrices of the Φ^(1,3) perturbations of the minimal unitary series. This analysis provides an identification of fields which create the massive kink spectrum. We investigate the ultraviolet limit of the restricted sine-Gordon model, and explain the relation between the restriction and the Fock space cohomology of minimal models. We also comment on the structure of degenerate vacuum states. Deformed Serre relations are proven for arbitrary affine Toda theories, and it is shown in certain cases how relations of the Serre type become fractional spin supersymmetry relations upon restriction.

Let ℋ be the corner-transfer-matrix Hamiltonian for the six-vertex model in the anti-ferroelectric regime. It acts on the infinite tensor product W=V⊗V⊗V⊗⋯, where V is the 2-dimensional irreducible representation of the quantum affine Lie algebra . We observe that ℋ is the derivation of , and conjecture that the eigenvectors of ℋ form the level-1 vacuum representation of . We report on checks in support of our conjecture.

An exactly solvable model of the pairing transition from a conductive to an insulator phase in a two-dimensional, log-potential Coulomb system is given. The system considered is the one-component plasma of positive particles without a neutralizing background near a metal interface, which is coupled to an external (non-Coulombic) potential. A complete analytic description of the equilibrium properties of the two phases and the transition is provided, and this is compared with the expected analytic properties of the two-component log-potential Coulomb gas in one and two-dimensions.

We use the Virasoro master equation to study the space of Lie h-invariant conformal field theories, which includes the standard rational conformal field theories as a small subspace. In a detailed example, we apply the general theory to characterize and study the Lie h-invariant graphs, which classify the Lie h-invariant conformal field theories of the diagonal ansatz on SO(n). The Lie characterization of these graphs is another aspect of the recently observed Lie group-theoretic structure of graph theory.

Path integrals with Hamiltonians on coadjoint orbits of SU(2) and SU(1,1) are explicitly calculated, taking the operator ordering into account. The path integral gives the kernel function of the unitary operator of the representation which is constructed by the method of Kirillov-Kostant theory.

We study the limit of the algebra associated with the solution of Z_n elliptic YBE. And we give its specific relation with the quantum algebra sl_q(n).

The Poisson structure of Lax operator of B and C type and super Lax operator which has odd parity are studied. The algebraic structure of Poisson structure as a background of Lax equation is thrown light on.

Generalized N=2 super KdV hierarchies are constructed based on super Lax equations associated with Lie superalgebras SL(n|n)⁽¹⁾. The equivalence of the scalar super Lax formalism and the Lie superalgebraic method is derived by taking account of gauge transformations regarding the centre of SL(n|n)⁽¹⁾. We show that generalized N=2 super KdV hierarchies are related to the even reductions of the super KP hierarchy.

We present N³-state R-matrix related with U_q(sl(3)) algebra at q^2N=1. Its matrix elements are interpreted as Boltzmann weights of an elementary box of some 2d lattice statistical model and given in terms of weights of the "minimal" sl(3) chiral Potts model.

This paper deals with various aspects of knot theory when fermionic degrees of freedom are taken into account in the braid group representations and in the state models. We discuss how the Ř matrix for the Alexander polynomial arises from the Fox differential calculus, and how it is related to the quantum group U_qgl(1,1). We investigate new families of solutions of the Yang Baxter equation obtained from "linear" representations of the braid group and exterior algebra. We study state models associated with U_qsl(n,m), and in the case n=m=1 a state model for the multivariable Alexander polynomial. We consider invariants of links in solid handlebodies and show how the non trivial topology lifts the boson fermion degeneracy that is present in S³. We use "gauge like" changes of basis to obtain invariants in thickened surfaces Σ×[0,1].