BRST cohomology of timelike Liouville theory

We compute the Hermitian sector of the relative BRST cohomology of the spacelike and timelike Liouville theories with generic real central charge cL in each case, coupled to a spacelike Coulomb gas and a generic transverse CFT. This paper is a companion of [1], and its main goal is to completely characterize the cohomology of the timelike theory with cL≤ 1 which was defined there. We also apply our formulas to revisit the BRST cohomology of the spacelike Liouville theory with cL> 1, which includes generalized minimal gravity. We prove a no-ghost theorem for the Hermitian sector in the timelike theory and for some spacelike models.