Virasoro Algebra, Vertex Operators, Quantum Sine-Gordon and Solvable Quantum Field Theories

Abstract

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).