The genus-zero bootstrap of chiral vertex operators in Liouville theory

Abstract

The F and B matrices associated with Virasoro null vectors are derived in closed form by making use of the operator-approach suggested by the Liouville theory. It is found that the entries of the fusing and braiding matrices are not simply equal to quantum-group symbols, but involve additional coupling constants. Their derivation neatly follows from the differential equations which express the decoupling of Virasoro null vectors, combined with the general schme of Moore and Seiberg. The present operator formalism, contrary to the Coulomb-gas approach, allows to study the coefficients of the operator-product algebra to all orders. Altogether the present work, and a subsequent article, give the cocrete realization of Moore and Seiberg's scheme that describes the chiral operator-algebra of two-dimensional gravity and minimal models.