New Developments in D-dimensional conformal quantum field theory

Abstract

A review of recent developments in conformal quantum field theory in D-dimensional space is presented. The conformally invariant solution of the Ward identities is studied. We demonstrate the existence of D-dimensional analogues of primary and secondary fields, the central charge, and the null vectors. The Hilbert space is shown to possess a specific model-independent structure defined by the 1 2 (D+1)(D+2) -dimensional symmetry and the Ward identities. In particular, there exists a sector H of the Hilbert space related to an infinite family of “secondary” fields which are generated by the currents and the energy-momentum tensor. The general solution of the Ward identities in D>2 defining the sector H necessarily includes the contribution of the gauge fields. We derive the conditions which single out the conformal theories of a direct (non-gauge) interaction. We examine the class of models satisfying these conditions. It is shown that the Green functions of the current and the energy-momentum tensor in these models are uniquely determined by the Ward identities for any D≥2. The anomalous Ward identities containing contributions of c-number and operator analogues of the central charge, are discussed. Closed sets of expressions for the Green functions of secondary fields are obtained in D-dimensional space. A family of exactly solvable conformal models in D≥2 is constructed. Each model is defined by the requirement of vanishing of a certain field Q s , s=1,2…. The fields Q s are constructed as definite superpositions of secondary fields. After that, one requires each field Q s to be primary. The latter is possible for specific values of scale dimensions of fundamental fields (a D-dimensional analogue of the Kac formula). The states Q s |0〉 are analogous to null vectors. One can derive closed sets of differential equations for higher Green functions in each of the models. These results are demonstrated on examples of several exactly solvable models in D>2. The approach developed here is based on the finite-dimensional conformal symmetry for any D≥2. However the family of models under consideration does have the structure identical to that of two-dimensional conformal theories. This analogy is discussed in detail. It is shown that when D=2, the above family coincides with the well-known family of models based on infinite-dimensional conformal symmetry. The analysis of this phenomenon indicates the possibility of existence of D-dimensional analogue of the Virasoro algebra.