SO(2, [formula omitted]) invariant ring structure of BRST cohomology and singular vectors in 2D gravity with c

Abstract

We consider BRST quantized 2D gravity coupled to conformal matter with arbitrary central charge c M= c( p, q)<1 in the conformal gauge. We apply a Lian-Zuckerman SO(2, C ) (( p, q) dependent) rotation to Witten's c M=1 chiral ground ring. We show that the ring structure generated by the (relative BRST cohomology) discrete states in the (matter⊗Liouville⊗ghosts) Fock module may be obtained by this rotation. We give also explicit formulae for the discrete states. For some of them we use new formulae for c<1 Fock modules singular vectors which we present in terms of Schur polynomials generalizing the c=1 expressions of Goldstone, while the rest of the discrete states we obtain by finding the proper SO(2, C ) rotation. Our formulae give the extra physical states (arising from the relative BRST cohomology) on the boundaries of the p× q rectangles of the conformal lattice and thus all such states in (1, q) or ( p, 1) models.