This chapter initiates the analysis of conformal field theories with Lie-algebraic symmetry, for which an affine Lie algebra ĝ arises as the spectrum-generating algebra. Such models are somewhat peculiar among conformal field theories in that they can be formulated directly in terms of an action. We will thus introduce them by means of this action and show how to extract from it their algebraic structure, which provides them with an alternative algebraic definition. Special emphasis is placed on the formulation of the concept of primary field, and its relation with the integrable representations of the affine algebra ĝ. A key construction along this program is that of the Sugawara energy-momentum tensor, which is presented in great detail. It leads directly to a differential equation for the correlation functions, the Knizhnik-Zamolodchikov equation, of which simple solutions are presented. In the second part of the chapter, we present various free-field representations. In addition to being extremely useful computational tools, they provide an illustration of the different concepts introduced in the first part.
KeywordsVertex Operator Conformal Block Conformal Field Theory Primary Field Vertex Representation
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