Remarks on the BRST-cohomology for $c_M > 1$ matter coupled to ``Liouville gravity"

Abstract

We describe the (chiral) BRST-cohomology of matter with central charge $1 0$) 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 $(xy)^n,\ n=0,1,2,\ldots$, and there are {\it 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 $r=\pm 1$, 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.