REMARKS ON THE BRST COHOMOLOGY FOR C(M)GREATER-THAN-1 MATTER COUPLED TO LIOUVILLE GRAVITY

We describe the (chiral) BRST cohomology of matter with central charge 1<c(M)<25 coupled to a "Liouville" theory, realized as a free field with a background charge Q(L) such that c(M)+c(L)=26. We consider two cases: (a) matter is realized by one free field with an imaginary background charge; (b) matter is realized by D free fields: c(M)=D. In case (a) the cohomology states can be labelled by integers r, s of a rotated c(M)=1 theory, but hermiticity imposes r=s. Thus there is still a discrete set of momenta p(M)(r,r),p(L)(r,r) such that there are non-trivial (relative) cohomology states at level r2 with ghost-numbers 0 or 1 (for r>0) and ghost-numbers 0 or -1 (for r<0). The (chiral) ground ring is isomorphic to a subring of the c(M)=1 theory which is (xy)n, n=0, 1, 2, ..., and there are no non-trivial currents acting on the ground ring. In case (b) there is no non-trivial relative cohomology for non-zero ghost-numbers and, for zero ghost-number, the cohomology groups are isomorphic to a (D-1)-dimensional on-shell "transverse" Fock space. The only exceptions are at level 1 for vanishing matter momentum and p(L)=Q(L)(1+r) with r= +/-1, where one has one more ghost-number zero and a ghost-number r cohomology state. All these results follow quite easily from the existing literature.