Singular vectors in logarithmic conformal field theories

Abstract

Null vectors are generalized to the case of indecomposable representations which are one of the main features of logarithmic conformal field theories. This is done by developing a compact formalism with the particular advantage that the stress energy tensor acting on Jordan cells of primary fields and their logarithmic partners can still be represented in form of linear differential operators. Since the existence of singular vectors is subject to much stronger constraints than in regular conformal field theory, they also provide a powerful tool for the classification of logarithmic conformal field theories. *

Logarithmic conformal field theory through nilpotent conformal dimensions

“…These results are consistent with those of [10]. We observe that LCFTs are possible only when the Kac determinant has multiple zeros.…”

“…These results are consistent with findings of [10]. However there is evidence that, this method does not give all singular vectors [23], perhaps conditions of equation (27) are too strong.…”

“…On the other hand in LCFT's the presence of a second null vector forces us to allow only certain values of c and ∆ as given by equation (83). We observe that some null vectors obtained in [10] are missing, in other words we have an incomplete set. The reason for this may be that equation (27) sets too strong a condition within LCFT [23].…”

“…Interesting work on this issues has been done by Flohr [9] showing that in the minimal series, if p and q take values which are not coprime and fall out of the allowed range, LCFTs result. Also, singular vectors in LCFTs were analyzed by Flohr [10], using grassman variables, which suggests that there is a connection with supersymmetry. Connection with supersymmetry has been discussed elsewhere too [11].…”

“…Indeed, if LCFTs are the natural framework for the AdS/CFT correspondence [15], then the natural supersymmetry of this theory also suggests that there is a possible connection between supersymmetry and LCFT. In this paper we follow up Flohr's [10] idea by considering a nilpotent variable θ, which is not anticommutative:…”