The Virasoro Algebra
In this chapter we shall study the Lie algebra Vect S 1 of vector fields on a circle and some of its generalizations. The Lie algebra Vect S 1 has a central extension, the Virasoro algebra. The representation theory of the Virasoro algebra is closely related to the representation theory of affine Lie algebras. In fact, through the Sugawara construction, to be defined below, a highest weight representation of an affine Lie algebra carries always a highest weight representation of the Virasoro algebra. All the irreducible highest weight representations of the Virasoro algebra are known and they can be exponentiated to representations of associated infinite-dimensional Lie groups. The representation theory of the algebra of vector fields on a higher dimensional manifold is much less understood; we shall discuss the extensions of these algebras in Section 7.6.
KeywordsCentral Charge High Weight Vector Virasoro Algebra High Weight Representation Finite Linear Combination
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