Torus Amplitudes in Minimal Liouville Gravity and Matrix Models

We evaluate one-point correlation numbers on the torus in the Liouville theory coupled to the conformal matter M(2,2p+1). We find agreement with the recent results obtained in the matrix model approach.Comment: 10 page

Frobenius manifolds, Integrable Hierarchies and Minimal Liouville Gravity

We use the connection between the Frobrenius manifold and the Douglas string equation to further investigate Minimal Liouville gravity. We search a solution of the Douglas string equation and simultaneously a proper transformation from the KdV to the Liouville frame which ensure the fulfilment of the conformal and fusion selection rules. We find that the desired solution of the string equation has explicit and simple form in the flat coordinates on the Frobenious manifold in the general case of (p,q) Minimal Liouville gravity.Comment: 17 pages; v2: typos removed, some comments added, minor correction

Higher Equations of Motion in N=2 Superconformal Liouville Field Theory

We present an infinite set of higher equations of motion in N=2 supersymmetric Liouville field theory. They are in one to one correspondence with the degenerate representations and are enumerated in addition to the U(1) charge \omega by the positive integers m or (m,n) respectively. We check that in the classical limit these equations hold as relations among the classical fields.Comment: 11 page

On the N=1 super Liouville four-point functions

We construct the four-point correlation functions containing the top component of the supermultiplet in the Neveu-Schwarz sector of the N=1 SUSY Liouville field theory. The construction is based on the recursive representation for the NS conformal blocks. We test our results in the case where one of the fields is degenerate with a singular vector on the level 3/2. In this case, the correlation function satisfies a third-order ordinary differential equation, which we derive. We numerically verify the crossing symmetry relations for the constructed correlation functions in the nondegenerate case.Comment: 23 page