Virasoro algebra

In mathematics, the Virasoro algebra is a complex Lie algebra and the unique nontrivial central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string theory. It is named after Miguel Ángel Virasoro.

Structure

The Virasoro algebra is spanned by generators for <math>n \isin \mathbb{Z}</math> and the central charge .

These generators satisfy <math>[c,L_n]=0</math> and

The factor of <math>\frac{1}{12}</math> 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 or Schottenloher, Thm. 5.1, pp. 79.

The Virasoro algebra has a presentation in terms of two generators (e.g. <sub>3</sub> and <sub>−2</sub>) and six relations.

The generators <math>L_{n>0}</math> are called annihilation modes, while <math>L_{n<0}</math> are creation modes. A basis of creation generators of the Virasoro algebra's universal enveloping algebra is the set

:<math>

\mathcal{L} = \Big\{ L_{-n_1} L_{-n_2} \cdots L_{-n_k}\Big\}_{\begin{array}{l} k\in\mathbb{N} \\ 0 < n_1 \leq n_2 \leq \cdots n_k\end{array

</math>

For <math>L\in \mathcal{L}</math>, let <math>|L|= \sum_{i=1}^k n_i</math>, then

<math>[L_0,L] = |L|L</math>.

Representation theory

In any indecomposable representation of the Virasoro algebra, the central generator <math>c</math> of the algebra takes a constant value, also denoted <math>c</math> and called the representation's central charge.

A vector <math>v</math> in a representation of the Virasoro algebra has conformal dimension (or conformal weight) <math>h</math> if it is an eigenvector of <math>L_0</math> with eigenvalue <math>h</math>:

: <math> L_0 v = hv</math>

An <math>L_0</math>-eigenvector <math>v</math> is called a primary state (of dimension <math>h</math>) if it is annihilated by the annihilation modes,

: <math> L_{n>0} v = 0</math>

Highest weight representations

A highest weight representation of the Virasoro algebra is a representation generated by a primary state <math>v</math>.

A highest weight representation is spanned by the <math>L_0</math>-eigenstates <math>\{Lv\}_{L\in\mathcal{L</math>. The conformal dimension of <math>Lv</math> is <math>h+|L|</math>, where <math>|L|\in\mathbb{N}</math> is called the level of <math>Lv</math>.

Any state whose level is not zero is called a descendant state of <math>v</math>.

For any <math>h,c\in\mathbb{C}</math>, the Verma module <math>\mathcal V_{c,h}</math> of central charge <math>c</math> and conformal dimension <math>h</math> is the representation whose basis is <math>\{Lv\}_{L\in\mathcal{L</math>, for <math>v</math> a primary state of dimension <math>h</math>.

The Verma module is

the largest possible highest weight representation.

The Verma module is indecomposable, and for generic values of <math>h,c\in\mathbb{C}</math> it is also irreducible. When it is reducible, there exist other highest weight representations with these values of <math>h,c\in\mathbb{C}</math>, called degenerate representations, which are quotients of the Verma module. In particular, the unique irreducible highest weight representation with these values of <math>h,c\in\mathbb{C}</math> 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 descendant and primary.

A sufficient condition for the Verma module <math>\mathcal V_{c,h}</math> to have a singular vector is <math>h=h_{r,s}(c)</math> for some <math>r,s\in\mathbb{N}^*</math>, where

:<math> h_{r,s}(c) = \frac14\Big( (\beta r - \beta^{-1}s)^2-(\beta-\beta^{-1})^2\Big)\ ,\quad \text{where} \quad c=1-6(\beta-\beta^{-1})^2\ . </math>

Then the singular vector has level <math>rs</math> and conformal dimension

:<math> h_{r,s}+rs = h_{r,-s} </math>

Here are the values of <math>h_{r,s}(c)</math> for <math>rs\leq 4</math>, together with the corresponding singular vectors, written as <math>L_{r,s}v</math> for <math>v</math> the primary state of <math>\mathcal{V}_{c,h_{r,s}(c)}</math>:

:<math>

\begin{array}{|c|c|l|}

\hline

r,s & h_{r,s} & L_{r,s}

\hline

1,1 & 0 & L_{-1}

\hline

2,1 & -\frac12 +\frac{3}{4} \beta^2 & L_{-1}^2 -\beta^2 L_{-2}

\hline

1,2 & -\frac12 + \frac{3}{4}\beta^{-2} & L_{-1}^2 -\beta^{-2} L_{-2}

\hline

3,1 & -1 +2 \beta^2 &L_{-1}^3 -4\beta^2 L_{-1}L_{-2}+2\beta^2(2\beta^2+1)L_{-3}

\hline

1,3 & -1 +2 \beta^{-2} &L_{-1}^3 -4\beta^{-2} L_{-1}L_{-2}+2\beta^{-2}(2\beta^{-2}+1)L_{-3}

\hline

4,1 & -\frac32 +\frac{15}{4} \beta^2 &

\begin{array}{r}

L_{-1}^4 -10\beta^2L_{-1}^2L_{-2}

+2\beta^2\left(12\beta^2+5\right) L_{-1}L_{-3}

+9\beta^4 L_{-2}^2

-6\beta^2\left(6\beta^4+4\beta^2+1\right)L_{-4}

\end{array}

\hline

2,2 & \frac34\left(\beta-\beta^{-1}\right)^2 &

\begin{array}{l}

L_{-1}^4-2\left(\beta^2+\beta^{-2}\right)L_{-1}^2L_{-2}

+\left(\beta^2-\beta^{-2}\right)^2 L_{-2}^2

+2\left(1+\left(\beta+\beta^{-1}\right)^2\right) L_{-1}L_{-3}

-2\left(\beta+\beta^{-1}\right)^2 L_{-4}

\end{array}

\hline

1,4 & -\frac32 +\frac{15}{4} \beta^{-2} &

\begin{array}{r}

L_{-1}^4 -10\beta^{-2}L_{-1}^2L_{-2}

+2\beta^{-2}\left(12\beta^{-2}+5\right) L_{-1}L_{-3}

+9\beta^{-4} L_{-2}^2

-6\beta^{-2}\left(6\beta^{-4}+4\beta^{-2}+1\right)L_{-4}

\end{array}

\hline

\end{array}

</math>

Singular vectors for arbitrary <math>r,s\in\mathbb{N}^*</math> may be computed using various algorithms, This can be further generalized to supermanifolds.

Vertex algebras and conformal algebras

The Virasoro algebra also has vertex algebraic and conformal algebraic counterparts, which basically come from arranging all the basis elements into generating series and working with single objects.

History

The Witt algebra (the Virasoro algebra without the central extension) was discovered by É. Cartan

</references>

Structure

Representation theory

Highest weight representations

Singular vectors

Vertex algebras and conformal algebras

History

Further reading