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The Coulomb-Gas Formalism

  • Philippe Di Francesco
  • Pierre Mathieu
  • David Sénéchal
Part of the Graduate Texts in Contemporary Physics book series (GTCP)

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 φφ + 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.

Keywords

Minimal Model Vertex Operator Conformal Block Conformal Dimension Fusion Rule 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York, Inc. 1997

Authors and Affiliations

  • Philippe Di Francesco
    • 1
  • Pierre Mathieu
    • 2
  • David Sénéchal
    • 3
  1. 1.Commissariat l’Énergie Atomique Centre d’Études de SaclayService de Physique ThéoriqueGif-sur-YvetteFrance
  2. 2.Département de PhysiqueUniversité LavalQuébecCanada
  3. 3.Département de PhysiqueUniversité de SherbrookeSherbrookeCanada

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