SO(2,C) invariant ring structure of BRST cohomology and singular vectors in 2D gravity with c < 1 matter

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 $SO(2,\bbc)$ ($(p,q)$ - 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 $\otimes$ Liouville $\otimes$ 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 $SO(2,\bbc)$ rotation. Our formulae give the extra physical states (arising from the relative BRST cohomology) on the boundaries of the $p\times q$ rectangles of the conformal lattice and thus all such states in $(1,q)$ or $(p,1)$ models.

15 pages

References in corpus (1)

• Interaction of Discrete States in Two-Dimensional String Theory

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