The Virasoro algebra has a presentation in terms of two generators (e.g. L3 and L−2) and six relations.
In any indecomposable representation of the Virasoro algebra, the central generator
of the algebra takes a constant value, also denoted
) if it is annihilated by the annihilation modes, A highest weight representation of the Virasoro algebra is a representation generated by a primary state
The Verma module is the largest possible highest weight representation.
The Verma module is indecomposable, and for generic values of
When it is reducible, there exist other highest weight representations with these values of
, called degenerate representations, which are quotients of the Verma module.
In particular, the unique irreducible highest weight representation with these values of
is the quotient of the Verma module by its maximal submodule.
It can be useful to use non-integer Kac indices for parametrizing the conformal dimensions of Verma modules that do not have singular vectors, for example in the critical random cluster model.
The inverse Shapovalov form is relevant to computing Virasoro conformal blocks, and can be determined in terms of singular vectors.
The representation is called unitary if that Hermitian form is positive definite.
Since any singular vector has zero norm, all unitary highest weight representations are irreducible.
An irreducible highest weight representation is unitary if and only if Daniel Friedan, Zongan Qiu, and Stephen Shenker showed that these conditions are necessary,[8] and Peter Goddard, Adrian Kent, and David Olive used the coset construction or GKO construction (identifying unitary representations of the Virasoro algebra within tensor products of unitary representations of affine Kac–Moody algebras) to show that they are sufficient.
of the Virasoro algebra is the function The character of the Verma module
is reducible due to the existence of a singular vector at level
This singular vector generates a submodule, which is isomorphic to the Verma module
This Verma module has an irreducible quotient by its largest nontrivial submodule.
(The spectrums of minimal models are built from such irreducible representations.)
The character of the irreducible quotient is This expression is an infinite sum because the submodules
Technically, the conformal bootstrap approach to two-dimensional CFT relies on Virasoro conformal blocks, special functions that include and generalize the characters of representations of the Virasoro algebra.
Since the Virasoro algebra comprises the generators of the conformal group of the worldsheet, the stress tensor in string theory obeys the commutation relations of (two copies of) the Virasoro algebra.
This is because the conformal group decomposes into separate diffeomorphisms of the forward and back lightcones.
Diffeomorphism invariance of the worldsheet implies additionally that the stress tensor vanishes.
Their theory is similar to that of the Virasoro algebra, now involving Grassmann numbers.
The Virasoro algebra is a central extension of the Lie algebra of meromorphic vector fields with two poles on a genus 0 Riemann surface.
On a higher-genus compact Riemann surface, the Lie algebra of meromorphic vector fields with two poles also has a central extension, which is a generalization of the Virasoro algebra.
The central extension of the Witt algebra that gives the Virasoro algebra was first found (in characteristic p > 0) by R. E. Block[13] (1966, page 381) and independently rediscovered (in characteristic 0) by I. M. Gelfand and Dmitry Fuchs[14] (1969).
The central extension giving the Virasoro algebra was rediscovered in physics shortly after by J. H. Weis, according to Brower and Thorn[16] (1971, footnote on page 167).