Masato Okado

Exactly Solvable Sos Models: Local Height Probabilities and Theta Function Identities

Abstract The local height probabilities (LHPs) of a series of solvable solid-on-solid (SOS) models are obtained. The models have been constructed by a fusion procedure from the eight-vertex SOS model. The LHP results are expressed in terms of modular functions (which we call the branching coefficients) appearing in appropriate theta function identities. The critical behavior is studied by using their automorphic properties. Some of these identities result from the representation theory of affine Lie algebras. The branching coefficients are (not necessarily irreducible) characters of the Virasoro algebra constructed from the pairs (A 1 (1) ⊗ A 1 (1) , A 1 (1) ) or (A 2 l −1 (1) , C l (1) ). As the critical exponents the present lattice models realize the anomalous dimensions of known 2D conformal field theories (the minimal unitarizable theory and its supersymmetric extensions, the Z N -invariant theory, etc.)

Solvable lattice models related to the vector representation of classical simple Lie algebras

A series of solvable lattice models with face interaction are introduced on the basis of the affine Lie algebraXn(1)=An(1),Bn(1),Cn(1),Dn(1). The local states taken on by the fluctuation variables are the dominant integral weights ofXn(1) of a fixed level. Adjacent local states are subject to a condition related to the vector representation ofXn. The Boltzmann weights are parametrized by elliptic theta functions and solve the star-triangle relation.

Paths, Crystals and Fermionic Formulae

We introduce a fermionic formula associated with any quantum affine algebra U q (X N (r) . Guided by the interplay between corner transfer matrix and the Bethe ansatz in solvable lattice models, we study several aspects related to representation theory, most crucially, the crystal basis theory. They include one-dimensional sums over both finite and semi-infinite paths, spinon character formulae, Lepowsky—Primc type conjectural formula for vacuum string functions, dilogarithm identities, Q-systems and their solution by characters of various classical subalgebras and so forth. The results expand [HKOTY1] including the twisted cases and more details on inhomogeneous paths consisting of non-perfect crystals. As a most intriguing example, certain inhomogeneous one-dimensional sums conjecturally give rise to branching functions of an integrable G 2 (1) -module related to the embedding G 2 (1) ↪ B 3 (1) ↪ D 4 1 .

Fusion of the eight vertex SOS model

A higher spin analogue is presented of the eight vertex-SOS correspondence in the sense of Andrews-Baxter-Forrester. The resulting hierarchy of solvable models contain the hard hexagon model and its recent multi-state generalizations.

Solvable lattice models whose states are dominant integral weights of A it?1 (1)

A new hierarchy of solvable IRF models is presented. It is generated from Belavins Zn×Znsymmetric model. The site variables take values in the set of level l dominant integral weights of A−1(1). It is conjectured that the local state probabilities are given through the irreducible decomposition of characters for the affine Lie algebra pair (An−1(1)⊕An−1(1),An−1(1)).

Combinatorics of representations of\(U_q (\widehat{\mathfrak{s}\mathfrak{l}}(n))\) atq=0

AbstractTheq=0 combinatorics for

ENERGY FUNCTIONS IN BOX BALL SYSTEMS

U_q (\widehat{\mathfrak{s}\mathfrak{l}}(n))

TheA n /(1) face models

is studied in connection with solvable lattice models. Crystal bases of highest weight representations of