METHOD OF SOLVING CONFORMAL MODELS IN D-DIMENSIONAL SPACE III SECONDARY FIELDS IN D>2 AND THE SOLUTION OF TWO-DIMENSIONAL MODELS

Abstract

We proceed with the study (started in Refs. 1 and 2) of the Hilbert space of conformal field theory in D di mensions. We discuss an infinite family of secondary fields [Formula: see text] generated by the action of the components of energy–momentum tensor Tμν on the fundamental (primary) field. It is shown that the states of these fields form a specific sector of the Hilbert space H which is determined by the Ward identities and [Formula: see text]-dimensional conformal symmetry. We demonstrate that for D = 2 the subspace H coincides with the space of representation of the Virasoro algebra. Each exactly solvable model in the case of D ≥ 2 is defined by the requirement of vanishing of a certain state Qs(x)|0> ⊂ H analogous to the null vector of two-dimensional theory. The Green functions of the fields [Formula: see text] are calculated in terms of the Green functions of the fundamental field. It is shown that all the Green functions of the type [Formula: see text] satisfy the anomalous Ward identities. The anomalous contributions are given by the fields [Formula: see text], where s′≤s-1. The fields Qs are constructed as superpositions of secondary fields with the anomalous contribution equal to zero, An approach developed is based on a finite-dimensional conformal symmetry for any D ≥ 2. Nevertheless the resulting models have the structure analogous to that of two-dimensional conformal theories. This analogy is discussed in detail. It is shown that for D = 2 the family of models coincides with the well-known family of conformal models based on infinite-dimensional conformal symmetry. The analysis of this phenomenon indicates the existence of the D-dimensional analog of the Virasoro algebra.