[2] A W-algebra is an associative algebra that is generated by the modes of a finite number of meromorphic fields
The rest of the commutation relations can in principle be determined by solving the Jacobi identities.
Most commonly studied W-algebras are freely generated, including the W(N) algebras.
[4] 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.
[3] Then the central charge of the W-algebra is a function of the level of the affine Lie algebra.
is the set of ordered tuples of strictly positive integers of the type
For generic values of the charges, the Verma module is the only highest weight representation.
If a Verma module is reducible, any indecomposible submodule is itself a highest weight representation, and is generated by a state that is both descendant and primary, called a null state or null vector.
If we start from a Verma module that has a maximal number of null vectors, and set all these null vectors to zero, we obtain an irreducible representation called a fully degenerate representation.
For example, in the case of the algebra W(3), the Verma module with vanishing charges
Setting these null vectors to zero yields a fully degenerate representation called the vacuum module.
The simplest nontrivial fully degenerate representation of W(3) has vanishing null vectors at levels 1, 2 and 3, whose expressions are explicitly known.
are functions of the momentum and the central charge, invariant under the action of the Weyl group.
The conformal dimension is[10] Let us parametrize the central charge in terms of a number
In a correlation function of primary fields, local Ward identities determine the action of
Global Ward identities further reduce the problem to determining three-point functions of the type
In the W(3) algebra, as in generic W-algebras, correlation functions of descendant fields can therefore not be deduced from correlation functions of primary fields using Ward identities, as was the case for the Virasoro algebra.
In the latter case, for a differential equation to exist, one of the other fields must have vanishing null vectors.
(almost fully degenerate) obeys a differential equation whose solutions are generalized hypergeometric functions of type
Their spaces of states are made of finitely many fully degenerate representations.
[12] For example, the two-dimensional critical three-state Potts model has central charge
, with one interaction term for each simple root: This depends on the cosmological constant
For the preservation of conformal symmetry in the quantum theory, it is crucial that there are no more interaction terms than components of the vector
Then it is possible to build a rational conformal field theory based on this W-algebra, which is logarithmic.
[15] The original definition, provided by Alexander Premet, starts with a pair
over the complex numbers and a nilpotent element e. By the Jacobson-Morozov theorem, e is part of a sl2 triple (e, h, f).
This induces a non-degenerate anti-symmetric bilinear form on the −1 graded piece by the rule: After choosing any Lagrangian subspace
, we may define the following nilpotent subalgebra which acts on the universal enveloping algebra by the adjoint action.