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 $1<c_M<25$ 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 $r^2$ 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 $(xy{)}^n,n=0,1,2,\dots$, 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.

10 pages

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