The Coulomb-Gas Formalism

Abstract

This chapter describes a representation of the conformal fields of minimal models in terms of vertex operators built from a free boson with special boundary conditions. This representation bears the name of Coulomb gas or modified Coulomb gas. This terminology comes from the resemblance of the free boson correla $\left\langle {\varphi (z,\bar z)} \right.\left. {\varphi (w,\bar w)} \right\rangle = - \ln {\left| {z - w} \right|^2}$ with the electric potential energy between two unit charges in two dimensions. In Sect. 9.1, we calculate the correlation function of vertex operators and indicate how the symmetry φ → φ + a of the boson theory imposes a constraint (the neutrality condition) on this correlation function. We then modify the free-boson action—or, equivalently, the energy-momentum tensor—and this modifies the central charge and the neutrality condition. This section is supplemented by App. 9.A, where the calculation of the modified energy-momentum tensor is detailed. In Sect. 9.2, we introduce the notion of screening operators and describe how the insertion of such operators in bosonic correlation functions allows for a sort of projection onto minimal-model correlation functions. Examples of correlation functions are calculated. Finally, in Sect. 9.3, we explain the general structure of the minimal-model correlation functions in this formalism. Special attention is devoted to the properties of conformal blocks, and the idea of a conformal field theory defined on a surface of arbitrary genus is introduced. The mathematical setting of the Coulomb-gas representation of minimal models (i.e., BRST cohomology of the bosonic Fock spaces) is described in App. 9.B.