Correlators, Null Vectors and Operator Algebra

Abstract

In the previous chapter, we have seen that null vectors play an essential role in the characterization of minimal conformal models. However, they also lead to more tangible consequences. In particular, one may derive from them linear partial differential equations the solution of which determines the four-point functions. Physical operators, however, have holomorphic and antiholomorphic contributions. These parts are connected because the coordinates z and z̄ are complex conjugates of each other. Physically sensible correlation functions of scalar operators (Δ = Δ̄) must be single-valued functions of z. Correlation functions of operators with non-trivial spins s = Δ− Δ̄ may pick up a phase under the exchange of two operators. However such a change must be consistent all over the possible physical n-point functions. This constitutes the monodromy problem for correlation functions and is the the subject of this chapter. Here, a particular property of two-dimensional conformal theories will become very helpful, namely the operator product algebra [619]. This is an extension of the short-distance expansion of the product of two operators to conformal theories. The particular features of the latter theories results in the existence of finite algebras of operators whose correlation functions can be exactly computed.