Êtes-vous un étudiant de l'EPFL à la recherche d'un projet de semestre?
Travaillez avec nous sur des projets en science des données et en visualisation, et déployez votre projet sous forme d'application sur Graph Search.
Concept
Algèbre de Virasoro
Résumé
In mathematics, the Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string theory.
The Virasoro algebra is spanned by generators Ln for n ∈ Z and the central charge c.
These generators satisfy and
The factor of is merely a matter of convention. For a derivation of the algebra as the unique central extension of the Witt algebra, see derivation of the Virasoro algebra.
The Virasoro algebra has a presentation in terms of two generators (e.g. L3 and L−2) and six relations.
A highest weight representation of the Virasoro algebra is a representation generated by a primary state: a vector such that
where the number h is called the conformal dimension or conformal weight of .
A highest weight representation is spanned by eigenstates of . The eigenvalues take the form , where the integer is called the level of the corresponding eigenstate.
More precisely, a highest weight representation is spanned by -eigenstates of the type with and , whose levels are . Any state whose level is not zero is called a descendant state of .
For any pair of complex numbers h and c, the Verma module is
the largest possible highest weight representation. (The same letter c is used for both the element c of the Virasoro algebra and its eigenvalue in a representation.)
The states with and form a basis of the Verma module. The Verma module is indecomposable, and for generic values of h and c it is also irreducible. When it is reducible, there exist other highest weight representations with these values of h and c, called degenerate representations, which are cosets of the Verma module. In particular, the unique irreducible highest weight representation with these values of h and c is the quotient of the Verma module by its maximal submodule.
A Verma module is irreducible if and only if it has no singular vectors.
A singular vector or null vector of a highest weight representation is a state that is both descendent and primary.
Source officielle
À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.