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LCFT and Sine-Liouville theory

LCFT is special conformal field theory with logarithmic terms in operator product expansion OPE. It is well-known that it is possible to construct operators with logarithmical terms in OPE at Liouville model. Here we discuss this topic for some generalization of Liouville model, such as Sine-Liouville model.


Introduction
Logarithmic conformal field theories LCFT are two-dimensional conformal field theories with logarithmic terms in operator product expansion OPE [1]. Logarithmic terms appear because of some primary operators have degeneration for conformal dimensions. Most famous example of such theory is c = −2 model. This model describes the system of ghosts with conformal dimensions 0 and 1. Also LCFT appear at WZWN model for some (super)groups and some levels k. Caux, Kogan and Tsvelik [2] had remarked that in Liouville model some primary operators generate LCFT. We will discuss LCFT for some generalization of Liouville model Sine-Liouville theory. Remarkable that Sine-Liouville theory [3][4] is dual to two-dimensional black hole (the Witten cigar) [5,6].
Let us consider WZWN model with SL(2, R) k /U (1)-coset in level k. The Witten solution [5,6] is described by the sigma-model with target metric ds 2 = k dr 2 + tanh 2 rdθ 2 . (1) This model has dilaton field Φ = log(cosh 2 r). Also this theory is CFT with central charge There is another parametrization of metric (1) Here spectrum of primary fields has form where m, n are integers and P is continuous.

Liouville and Sine-Liouville theories
The Liouville model has the following action where integration is carried out over some Riemann surface with metric g and the Ricci scalar R (2) , µ is a cosmological constant and b is a coupling constant. For simplicity we will discuss case of disk with R (2) = 0. For conformal invariance is necessary to Here the energy-momentum tensor is and the central charge The primary field : e αϕ : has a dimension The Sine-Liouville theory on disk has the following action The field φ is sum of φ L and φ R . Here the stress-energy tensor is For matching of the parameters for both models [4] we need coincidence of central charges Matching of another parameters is given by Also there is requirement that last term in action has conformal dimension 1

Logarithmical conformal field theories
Logarithmical terms [1,2,7] appear in operator product expansion when L 0 component of stressenergy tensor acts by mixing the primary fields. Simplest example with two fields is given by The fields C and D form Jordan cell. The operators C and D for Lioville theory were presented in [2]. Let's consider primary field V α = : e αϕ(z,z) := C, where α = Q/2. There is another operator with the same conformal dimension ∆ = Q 2 /8 Here operator D can be find fromṼ α D = 2 α : ϕe αϕ(z,z) : .

LCFT in Sine-Liouville theory
Let us consider primary fields of Sine-Lioville model [6] Ψ α,n,m =: Here right and left dimensions of these operators are ∆ ± α,n,m = α Matching with spectrum of SL(2, R) k /U (1) model (3) if one put iP = α − 1 4b . Let Ψ α,n,m be C operator. Then like Liouville theory D operator is There are not another operators with logarithmical terms in OPE because of stress-energy tensor (10).