A pseudo–conformal representation of the Virasoro algebra

Generalizing the concept of primary fields, we find a new representation of the Virasoro algebra, which we call it a pseudo–conformal representation. In special cases, this representation reduces to ordinary– or logarithmic–conformal field theory. There are, however, other cases in which the Green functions differ from those of ordinary– or logarithmic–conformal field theories. This representation is parametrized by two matrices. We classify these two matrices, and calculate some of the correlators for a simple example. In an ordinary conformal field theory primary fields are the highest weights of the representations of the Virasoro algebra. A primary field φ(w, w̄) can be defined through its operator product expansion with the stress-energy tensor T (z) ( and T̄ (z̄) ) or equivalently through its commutation relations with the Laurent expansion coefficients of T ; Ln’s [1]: [Ln, φi(z)] = z ∂zφi + (n+ 1)z ∆iφi, (1) where ∆i is the conformal weight of φi. One can regard ∆i’s as the diagonal elements of a diagonal matrix D, [Ln, φi(z)] = z ∂zφi + (n+ 1)z Dijφj . (2) One can however, extend the above relation for any matrix D, which is not necessarily diagonal. This new representation of Ln also satisfies the Virasoro algebra for any arbitrary matrix D [2]. By a suitable change of basis, one can make D diagonal or Jordanian. If it is diagonalizable, the field theory is nothing but the ordinary conformal field theory (CFT). Otherwise it should be in the Jordanian form. The latter case is the logarithmic conformal field theory (LCFT) [3, 4, 2]. In the simplest case, the Jordanian block is two dimensional and the relation (2) for the two fields φ and ψ, becomes [Ln, φ(z)] = z ∂zφ+ (n+ 1)z ∆φ [Ln, ψ(z)] = z ∂zψ + (n+ 1)z ∆ψ + (n+ 1)zφ. (3) The field φ is an ordinary primary field, and the field ψ is called a quasi-primary or logarithmic field and they transform in the following way: φ(z) → ( −1 ∂z )φ(f(z)) ψ(z) → ( −1 ∂z )[ψ(f(z)) + log( (z) ∂z )φ(f(z)] (4) The two-point functions of these fields has been obtained in [3, 4]. It has been shown in [2] that any n−point function ( for n > 2 ) containing the field ψ can be obtained through the n−point function containing the field φ instead of ψ. Now the natural question which may arise is that, “is it possible to generalize (2) such that Ln’s are still a representation of Virasoro algebra?” To investigate this question, we consider the following generalization of equation (2) [Ln, φi(z)] = z Bij∂zφj + (n+ 1)z Aijφj + Ci, (5) and impose the condition that Ln’s satisfy the Virasoro algebra: [[Ln, Lm], φi] = (n−m)[Lm+n, φi]. (6) Now it is easy to see that the generalization (5) satisfies the Virasoro algebra provided that the matrices A,B and C satisfy the following relations: B +BA− AB = B (7) BA = A (8) C = 0 (9) Now we try to classify the solutions of A and B. The trivial solution is B = 1, which is nothing but CFT, when A is diagonizable, and LCFT, when A is not diagonizable. If A is an invertible matrix the only solution for B is B = 1. However for any CFT which contains identity or any other field with zero conformal weight, A is not invertible, and there exists a corresponding new theory with B 6= 1 for which Ln’s satisfy the Virasoro algebra. This is obviously not a CFT any more, as the action of any diffeomorphism on a field contains a term −ξ · ∂φ, where ξ is the generator of the diffeomorphism. This