We study a family of exactly solvable models of conformally invariant quantum field theory in D-dimensional space. We demonstrate the existence of D-dimensional analogs of primary and secondary fields. Under the action of the energy–momentum tensor and conserved currents, the primary field creates an infinite set of (tensor) secondary fields of different generations. The commutators of secondary fields with zero components of the current and energy–momentum tensor include anomalous operator terms. We show that the Hilbert space of conformal theory has a special sector whose structure is solely defined by the Ward identities independently of the choice of dynamical model. The states of this sector are constructed from secondary fields. Definite self-consistent conditions on the states of the latter sector fix the choice of the field model uniquely. The above self-consistent conditions are formulated as follows. Special superpositions Qs, s= 1, 2, …, of secondary fields are constructed. Each superposition is determined by the requirement that the form of its commutators with energy–momentum tensor of a primary fiel d. Each equation Qs(x)=0 is consistent, and defines an exactly solvable model for D ≥ 3. The structure of these models is analogous to that of well-known two-dimensional conformal models. The states Qs(x)|0> are analogous to the null vectors of two-dimensional theory. In each of these models one can obtain a closed set of differential equations for all the higher Green functions, as well as algebraic equations relating the scale dimension of the fundamental field to the D-dimensional analog of a central charge. As an example, we present a detailed discussion on a pair of exactly solvable models in even-dimensional space D≥4.
Exactly solvable models of two-dimensional dilaton gravity and quantum eternal black holes
