In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Virasoro algebra (named after the physicist
Miguel Ángel Virasoro) is a complex
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
and the unique
central extension of the
Witt algebra
In mathematics, the complex Witt algebra, named after Ernst Witt, is the Lie algebra of meromorphic vector fields defined on the Riemann sphere that are holomorphic except at two fixed points. It is also the complexification of the Lie algebra ...
. It is widely used in
two-dimensional conformal field theory
A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations.
In contrast to other types of conformal field theories, two-dimensional conformal fie ...
and in
string theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...
.
Definition
The Virasoro algebra is
spanned by generators for and the
central charge
In theoretical physics, a central charge is an operator ''Z'' that commutes with all the other symmetry operators. The adjective "central" refers to the center of the symmetry group—the subgroup of elements that commute with all other element ...
.
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
In mathematics, the complex Witt algebra, named after Ernst Witt, is the Lie algebra of meromorphic vector fields defined on the Riemann sphere that are holomorphic except at two fixed points. It is also the complexification of the Lie algebra ...
, see
derivation of the Virasoro algebra.
The Virasoro algebra has a
presentation
A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
in terms of two generators (e.g.
3 and
−2) and six relations.
Representation theory
Highest weight representations
A
highest weight representation In the mathematical field of representation theory, a weight of an algebra ''A'' over a field F is an algebra homomorphism from ''A'' to F, or equivalently, a one-dimensional representation of ''A'' over F. It is the algebra analogue of a multiplic ...
of the Virasoro algebra is a representation generated by a primary state: a vector
such that
:
where the number is called the conformal dimension or conformal weight of
.
[P. Di Francesco, P. Mathieu, and D. Sénéchal, ''Conformal Field Theory'', 1997, .]
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 and , the
Verma module Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics.
Verma modules can be used in the classification of irreducible representations of a complex semisimple Lie algebra. Spe ...
is
the largest possible highest weight representation. (The same letter is used for both the element 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 and it is also irreducible. When it is reducible, there exist other highest weight representations with these values of and , called degenerate representations, which are cosets of the Verma module. In particular, the unique irreducible highest weight representation with these values of and 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.
Singular vectors
A singular vector or null vector of a highest weight representation is a state that is both descendent and primary.
A sufficient condition for the Verma module
to have a singular vector at the level
is
for some positive integers
such that
, with
:
In particular,
, and the reducible Verma module
has a singular vector
at the level
. Then
, and the corresponding reducible Verma module has a singular vector
at the level
.
This condition for the existence of a singular vector at the level
is not necessary. In particular, there is a singular vector at the level
if
with
and
. This singular vector is now a descendent of another singular vector at the level
. This type of singular vectors can however only exist if the central charge is of the type
:
.
(For
coprime, these are the central charges of the
minimal models.)
Hermitian form and unitarity
A highest weight representation with a real value of
has a unique
Hermitian form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allow ...
such that the Hermitian adjoint of
is
and the norm of the primary state is one.
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.
The
Gram determinant of a basis of the level
is given by the Kac determinant formula,
:
where the function ''p''(''N'') is the
partition function, and
is a positive constant that does not depend on
or
.
The Kac determinant formula was stated by
V. Kac (1978), and its first published proof was given by Feigin and Fuks (1984).
The irreducible highest weight representation with values and is unitary if and only if either ≥ 1 and ≥ 0, or
:
and ''h'' is one of the values
:
for ''r'' = 1, 2, 3, ..., ''m'' − 1 and ''s'' = 1, 2, 3, ..., ''r''.
Daniel Friedan
Daniel Harry Friedan (born October 3, 1948) is an American theoretical physicist and one of three children of the feminist author and activist Betty Friedan. He is a professor at Rutgers University.
Biography Education and career
Friedan earned h ...
, Zongan Qiu, and
Stephen Shenker (1984) showed that these conditions are necessary, and
Peter Goddard, Adrian Kent, and
David Olive (1986) used the
coset construction
In mathematics, the coset construction (or GKO construction) is a method of constructing unitary highest weight representations of the Virasoro algebra, introduced by Peter Goddard, Adrian Kent and David Olive (1986). The construction produce ...
or
GKO construction
In mathematics, the coset construction (or GKO construction) is a method of constructing unitary highest weight representations of the Virasoro algebra, introduced by Peter Goddard, Adrian Kent and David Olive (1986). The construction produces ...
(identifying unitary representations of the Virasoro algebra within tensor products of unitary representations of affine
Kac–Moody algebra
In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently and simultaneously discovered them in 1968) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a g ...
s) to show that they are sufficient.
Characters
The character of a representation
of the Virasoro algebra is the function
:
The character of the Verma module
is
:
where
is the
Dedekind eta function
In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string ...
.
For any
and for
, 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
. The quotient of
by this submodule is irreducible if
does not have other singular vectors, and its character is
:
Let
with