Logarithmic Correlation Functions in Liouville Field Theory

We study four-point correlation functions with logarithmic behaviour in Liouville field theory on a sphere, which consist of one kind of the local operators. We study them as non-integrated correlation functions of the gravitational sector of two-dimensional quantum gravity coupled to an ordinary conformal field theory in the conformal gauge. We also examine, in the (p, q) minimal conformal field theories, a condition of the appearance of logarithmic correlation functions of gravitationally dressed operators. ∗ E-mail: [email protected] One of physical examples in logarithmic conformal field theories [1]–[3] is the gravitationally dressed conformal field theory. Two-dimensional quantum gravity coupled to a free massless Majorana fermion field theory in the light-cone gauge had been studied in Ref. [4], where the non-integrated four-point correlation function of the gravitationally dressed operators, which has logarithmic behaviour, was obtained. The origin of the logarithms of such models is Liouville field theory which describes the gravitational sector of the system. However, correlation functions with logarithmic behaviour in the Liouville field theory are not sufficiently understood so far. The purpose of this letter is to study four-point correlation functions with logarithmic behaviour in Liouville field theory on a sphere. We consider a certain class of four-point correlation functions of local Liouville operators, which can be regarded as non-integrated correlation functions of the gravitational sector of two-dimensional quantum gravity coupled to an ordinary conformal field theory in the conformal gauge. We obtain the correlation functions with logarithmic behaviour and find the appearance of logarithms is the same mechanism as that in the Coulomb-gas construction of the correlation function in the c = −2 conformal field theory [5]. We also examine a condition of the appearance of logarithmic correlation functions consist of one kind of gravitationally dressed operators in the (p, q) minimal conformal field theories, and obtain the correlation function. The appearance of the correlation function with logarithmic behaviour in the gravitational dressing of the Majorana fermion field theory [4] can be also understood in our formulation. We first consider two-dimensional quantum gravity coupled to a conformal field theory on a sphere. The conformal field theory can be regarded as the matter part in the system. After conformal gauge fixing, the gravitational sector is described by the Liouville field theory with the action [6] SL[ĝ, φ] = 1 8π ∫