We compute the eigenvalues of the 3-state superintegrable chiral Potts model and of the associated spin chain by use of a functional equation. We find that the system has four phases, two of which are massless and two of which are massive.

After a brief review of the history of the star-triangle equation, we shall illustrate its importance with a few results for the two-dimensional Ising model and its generalization to the chiral Potts model. We shall discuss how the recent solutions in terms of higher-genus Riemann surfaces emerge. We shall finish with some further results for the quantities of interest in these new models. More related work is presented in the talks by Profs. Baxter and McCoy.

We consider conformal field theories on manifolds with a boundary, and the constraints placed by modular invariance on their partition functions. In particular, the partition functions on an annulus with particular boundary conditions are given by the fusion rules. This leads to a simple derivation of the Verlinde formula. We note the remarkable fact that, for some integrable models, these partition functions have the same form away from criticality, with the modular parameter $q$ of the annulus replaced by a temperature-like variable, and give a partial explanation of this in the case of the Ising model.

A general theory is presented to construct representations of the braid group and link polynomials (topological invariants for knots and links) from exactly solvable models in statistical mechanics at criticality. Sufficient conditions for the existence of the Markov trace are explicitly shown. Application of the theory to IRF and vertex models yields various link polynomials including an infinite sequence of new invariants. The new link polynomials are extended into two-variable link invariants. For the models with crossing symmetry, braid-monoid algebras associated with the link polynomials are derived. It is found that the Yang–Baxter relation gives both an algebraic approach and a graphical approach in knot theory.

We introduce a formal group naturally associated with algebraic curves. The formal group is isomorphic to the one obtained from the universal Witt scheme. The charge zero sector of the boson Fock space is regarded as the coordinate ring of the formal group. Using this structure, we can give tau functions. We also define new operators $f_n$, $v_n$ ($n \in \mathbf{Z}, n > 0$) on the fermion Fock space.

Presented is a new family of solvable solid-on-solid models in two-dimensional statistical mechanics. The site variables take the values in a set of not necessarily dominant integral weights of the affine Lie algebra $A^{(1)}_n$. The local state probabilities are obtained and the critical behavior is studied. Our family gives an extension of some recently discovered hierarchies of solvable solid-on-solid models.

The Yang–Baxter algebras (YBA) are introduced and formulated in a general way stressing graphical methods. Their various physical applications are then exposed: lattice statistical models, integrable field theories and factorizable S-matrices. The Bethe Ansatz (BA) and its generalizations provide the explicit solutions of all these models using the appropiate YBA. The six-vertex model solution is exposed. YB algebras and their associated physical models are classified in terms of simple Lie algebras.

It is exposed how these lattice models yield both solvable massive QFT and conformal models in appropiated scaling (continuous) limits within the lattice light-cone approach.

The method of finite-size calculations from the BA is exposed as well as its applications to derive the conformal properties of integrable lattice models. It is conjectured that all integrable QFT and conformal models follow in a scaling limit from these YB algebras.

To conclude braid and quantum groups are derived from the YBA in the limit of infinite spectral parameter.

Equations of motion for two-dimensional quantum field theory obtained as some relevant perturbation around CFT are analyzed. It is shown that for particular degenerate fields taken as the perturbations, the resulting field theories posseses non-trivial local integrals of motion. The example is the scaling limit of the Ising model at $T = T_c$ but with non-vanishing magnetic field. Implications of the integrals of motion for corresponding particle theory are discussed and the exact S-matrix for the critical Ising model with magnetic field is conjectured.

We give corrections or comments to the five points in our paper [1].