Operator products in 2-dimensional critical theories

Abstract

A noteworthy feature of certain conformally invariant 2-dimensional theories, such as the Ising and 3-state Potts models at the critical point, is the existence of “degenerate primary fields” associated with nullvectors of the Virasoro algebra. Such fields are endowed with a remarkably simple multiplication table under the operator product expansion, known as the fusion rules. In addition, correlation functions made up of these fields satisfy a system of linear homogeneous partial differential equations. We show here that these two properties are intimately related: for any n-point function, the number of conformally invariant solutions to the system of equations equals the number of times that the identity operator appears in the fusion of all n fields in the correlator. This theorem permits the calculation of some apparently intractable correlation functions. Finally, we generalize these ideas to the Neveu-Schwarz sector of superconformal theories.