Correlation Functions of (2k−1, 2) Minimal Matter Coupled to 2D Quantum Gravity

We compute N -point correlation functions of non-unitary (2k−1, 2) minimal matter coupled to 2D quantum gravity on a sphere using the continuum Liouville field approach. A gravitational dressing of the matter primary field with the minimum conformal weight is used as the cosmological operator. Our results are in agreement with the correlation functions of the one-matrix model at the k-th critical point. Recent remarkable progress of 2D quantum gravity has been made by the matrix model approach. [1,2] Matrix models provided us important and interesting understanding of non-perturbative aspects of 2D quantum gravity. Another approach called the Liouville approach [3,4] uses a continuum field theory and also has been extensively studied. [5−12] From the view point of the Liouville approach, it is important to compute correlation functions in order to study the non-trivial Liouville dynamics and in order to understand the precise connection with the matrix models. By using a free field approach and an analytic continuation procedure, N -point correlation functions for general N without the screening charges in c ≤ 1 conformal matters coupled to 2D gravity have been computed and analyzed in Refs. 10−12. They can be interpreted as scattering amplitudes of critical string theories in 2D target space with non-trivial background fields. In the case of c < 1 minimal matters [13,14] which require the screening charges, three-point functions have been computed. [6,11] The results in unitary minimal matters were found to be consistent with the matrix model results. In the Liouville approach, correlation functions of non-unitary minimal matters are not sufficiently understood so far. By comparing the gravitational dimensions [3,4] of physical operators in the Liouville approach and the matrix model approach, one finds that a gravitational dressing of the matter primary field with the minimum conformal weight should be used as the cosmological operator. [5,9] For unitary matters the cosmological operator is the ordinary one depending only on the Liouville field since the matter minimum weight primary field is the identity operator. On the other hand, for non-unitary minimal matters one must use a modified cosmological operator which depends on both of the matter and the Liouville fields. In Ref. 15 three-point functions of non-unitary (2k−1, 2) minimal matter [13] were computed. However, two kinds of gravitational dressings of the matter identity operator were used as cosmological operators as in Ref. 8. It is not clear to us whether such a method is appropriate to non-unitary matters. The purpose of this letter is to compute N -point functions of the non-unitary (2k−1, 2) minimal matter coupled to 2D gravity on a sphere. They are obtained in the Liouville approach using the action with the modified cosmological operator. We use a similar technique as that used in Ref. 11 to compute N -point functions without screening charges. Our results are in agreement with the correlation functions of the one-matrix models at the k-th critical point, [2] which are believed to represent the (2k−1, 2) minimal matter coupled to 2D gravity. Recently, we received a preprint by Govindarajan et al., [16] in which N -point functions of the (2k−1, 2) matter were obtained using different method from ours. They used algebraic properties of the BRST cohomology of the theory. Our results are also consistent with theirs. We now consider the non-unitary (2k−1, 2) minimal matter [13] coupled to 2D quantum gravity. The matter conformal field theory has the central charge 2 c = 1− 12α 0 , (1) with α0 = − 1 2 α+ + 1 2 α− , α+ = − 2 √ 2k − 1 , α− = − √ 2k − 1 . (2) After conformal gauge fixing, the system is described by the matter field X and the Liouville field φ with the action [4,14]