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Concept
Virasoro algebra
Summary
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.
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