Remarks on the BRST cohomology for cM > 1 matter coupled to “Liouville gravity”

Abstract

Listen

Translate article

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 (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.