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

TL;DR

This paper analyzes the structure of BRST cohomology in 2D gravity with matter of central charge less than one, using an $SO(2,bc)$ rotation to relate it to the known $c=1$ case and providing explicit formulas for discrete states.

Contribution

It introduces a method to derive the ring structure of BRST cohomology for $c<1$ models via an $SO(2,bc)$ rotation from the $c=1$ case and presents new formulas for singular vectors in Fock modules.

Findings

Explicit formulas for discrete states in $c<1$ models.

Demonstration of the $SO(2,bc)$ rotation relating $c<1$ and $c=1$ cohomologies.

Identification of physical states on the boundaries of the conformal lattice.

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.