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

TL;DR

This paper analyzes the BRST-cohomology of matter coupled to Liouville gravity for central charges between 1 and 25, revealing structure differences depending on matter realization and identifying the ground ring and cohomology states.

Contribution

It provides a detailed description of the BRST-cohomology structure for matter with $c_M > 1$ coupled to Liouville theory, including cases with imaginary background charge and multiple free fields.

Findings

Cohomology states labeled by integers with hermiticity constraints.

Ground ring isomorphic to a subring of the $c_M=1$ theory.

No non-trivial currents acting on the ground ring.

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 $(xy)^n,\ n=0,1,2,\ldots$, 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.