States of non-zero ghost number in $c<1$ matter coupled to 2d gravity

TL;DR

This paper explores the structure of non-zero ghost number states in $c<1$ matter coupled to 2D gravity, revealing their relation to primary states via descent equations and analyzing the Virasoro structure of the Liouville space.

Contribution

It introduces a method to construct non-zero ghost number states using double cohomology and descent equations in the Coulomb gas formalism.

Findings

States outside the primary grid are related to non-zero ghost number states.

Descent equations enable construction of non-zero ghost number states.

Correlation functions involving these states can be mapped to ghost number conserving ones.

Abstract

We study $c<1$ matter coupled to gravity in the Coulomb gas formalism using the double cohomology of the string BRST and Felder BRST charges. We find that states outside the primary conformal grid are related to the states of non-zero ghost number by means of descent equations given by the double cohomology. Some aspects of the Virasoro structure of the Liouville Fock space are studied. As a consequence, states of non-zero ghost number are easily constructed by ``solving'' these descent equations. This enables us to map ghost number conserving correlation functions involving non-zero ghost number states into those involving states outside the primary conformal grid.