Method of solving conformal models in D-dimensional space 3: Secondary fields in D > 2 and the solution of two dimensional models

摘要

We proceed with the study of the Hilbert space of conformal field theory in D dimensions. We discuss an infinite family of secondary fields P(sup T)(sub s) generated by the action of the components of energy-momentum tensor T(sub (mu)(nu)) 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 1/2 (D+1) (D+2)-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 Q(sub s)(x) modul 0 > is an element of H analogous to null-vector of two dimensional theory. The Green functions of the fields P(sup T)(sub s) are calculated in terms of the Green functions of fundamental field. It is shown that all the Green functions of the type satisfy the anomalous Ward identities. The anomalous contributions are given by the fields P(sup T)(sub s'), where s' (<=) s - 1. The fields Q(sub s) are constructed as superpositions of secondary fields with the anomalous contribution equal to zero, i.e. having the transformation properties of primary fields. 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 this family of fields 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. (author). 17 refs. (Atomindex citation 27:066687)