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Concept
W-algebra
Summary
In conformal field theory and representation theory, a W-algebra is an associative algebra that generalizes the Virasoro algebra. W-algebras were introduced by Alexander Zamolodchikov, and the name "W-algebra" comes from the fact that Zamolodchikov used the letter W for one of the elements of one of his examples.
A W-algebra is an associative algebra that is generated by the modes of a finite number of meromorphic fields , including the energy-momentum tensor . For , is a primary field of conformal dimension . The generators of the algebra are related to the meromorphic fields by the mode expansions
The commutation relations of are given by the Virasoro algebra, which is parameterized by a central charge . This number is also called the central charge of the W-algebra. The commutation relations
are equivalent to the assumption that is a primary field of dimension .
The rest of the commutation relations can in principle be determined by solving the Jacobi identities.
Given a finite set of conformal dimensions (not necessarily all distinct), the number of W-algebras generated by may be zero, one or more. The resulting W-algebras may exist for all , or only for some specific values of the central charge.
A W-algebra is called freely generated if its generators obey no other relations than the commutation relations. Most commonly studied W-algebras are freely generated, including the W(N) algebras. In this article, the sections on representation theory and correlation functions apply to freely generated W-algebras.
While it is possible to construct W-algebras by assuming the existence of a number of meromorphic fields and solving the Jacobi identities, there also exist systematic constructions of families of W-algebras.
From a finite-dimensional Lie algebra , together with an embedding , a W-algebra may be constructed from the universal enveloping algebra of the affine Lie algebra by a kind of BRST construction.
Then the central charge of the W-algebra is a function of the level of the affine Lie algebra.
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