Fermion variables are constructed for the 3D Ising Model in which it becomes a theory of free fermion strings defined on lattice.

We present the vertex operator construction of null fields (singular vertex operators) based on the oscillator representation of conformal and superconformal algebras with central charge extention. As a consequence of the null field construction we determine the degenerate highest weights and prove the corresponding Kac determinant formulae for the conformal ($N=0$) and superconformal ($N=1, 2$) algebras with $N$ supercharges. We also present an explicit construction of spin fields in the $N=1$ oscillator representations of the Neveu–Schwarz and Ramond algebras with extended central charges.

The relationship between the conformal field theories and the soliton equations (KdV, MKdV and Sine–Gordon, etc.) at both quantum and classical levels is discussed. The quantum Sine–Gordon theory is formulated canonically. Its Hamiltonian is the vertex operator with respect to the Feigin–Fuchs–Miura form of the Virasoro algebra with central charge $c\le1$. It is found that the quantum conserved quantities of the Sine–Gordon-MKdV hierarchy are expressed as polynomial functions of the Virasoro generators. In other words, an infinite set of mutually commutative polynomial functions of the Virasoro generators is obtained. A very simple recursion formula for the quantum conserved quantities is found for the special case of $\beta^2_c=8\pi$ ($\beta_c$ is the coupling constant in Coleman’s theory of quantum Sine–Gordon).

A supersymmetric extension of the Kadomtsev–Petviashvili hierarchy (SKP hierarchy) is proposed. It is shown that the solution space is identical with the universal super Grassmann manifold. An explicit formula of a solution to the SKP hierarchy is presented as a quotient of superdeterminants. Our arguments are based upon algebraic investigations of the structure of a linear supersymmetric differential equation.