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.
Definition A W-algebra is an associative algebra that is generated by the modes of a finite number of meromorphic fields W ( h ) ( z ) {\displaystyle W^{(h)}(z)} , including the energy-momentum tensor T ( z ) = W ( 2 ) ( z ) {\displaystyle T(z)=W^{(2)}(z)} . For h ≠ 2 {\displaystyle h\neq 2} , W ( h ) ( z ) {\displaystyle W^{(h)}(z)} is a primary field of conformal dimension h ∈ 1 2 N ∗ {\displaystyle h\in {\frac {1}{2}}\mathbb {N} ^{*}} . The generators ( W n ( h ) ) n ∈ Z {\displaystyle (W_{n}^{(h)})_{n\in \mathbb {Z} }} of the algebra are related to the meromorphic fields by the mode expansions
W ( h ) ( z ) = ∑ n ∈ Z W n ( h ) z − n − h {\displaystyle W^{(h)}(z)=\sum _{n\in \mathbb {Z} }W_{n}^{(h)}z^{-n-h}}
The commutation relations of L n = W n ( 2 ) {\displaystyle L_{n}=W_{n}^{(2)}} are given by the Virasoro algebra, which is parameterized by a central charge c ∈ C {\displaystyle c\in \mathbb {C} } . This number is also called the central charge of the W-algebra. The commutation relations
[ L m , W n ( h ) ] = ( ( h − 1 ) m − n ) W m + n ( h ) {\displaystyle [L_{m},W_{n}^{(h)}]=((h-1)m-n)W_{m+n}^{(h)}}
are equivalent to the assumption that W ( h ) ( z ) {\displaystyle W^{(h)}(z)} is a primary field of dimension h {\displaystyle h} . The rest of the commutation relations can in principle be determined by solving the Jacobi identities. Given a finite set of conformal dimensions H {\displaystyle H} (not necessarily all distinct), the number of W-algebras generated by ( W ( h ) ) h ∈ H {\displaystyle (W^{(h)})_{h\in H}} may be zero, one or more. The resulting W-algebras may exist for all c ∈ C {\displaystyle c\in \mathbb {C} } , 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.
Constructions While it is possible to construct W-algebras by assuming the existence of a number of meromorphic fields W ( h ) ( z ) {\displaystyle W^{(h)}(z)} and solving the Jacobi identities, there also exist systematic constructions of families of W-algebras.
Drinfeld-Sokolov reduction From a finite-dimensional Lie algebra g {\displaystyle {\mathfrak {g}}} , together with an embedding s l 2 ↪ g {\displaystyle {\mathfrak {sl}}_{2}\hookrightarrow {\mathfrak {g}}} , a W-algebra may be constructed from the universal enveloping algebra of the affine Lie algebra g ^ {\displaystyle {\hat {\mathfrak {g}}}} 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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