Virasoro algebra

A lowest weight representation is spanned by eigenstates of $L_{0}$ . The eigenvalues take the form $h+N$ , where the integer $N\geq 0$ is called the level of the corresponding eigenstate.

More precisely, a lowest weight representation is spanned by $L_{0}$ -eigenstates of the type $L_{-n_{1}}L_{-n_{2}}\cdots L_{-n_{k}}v$ with $0<n_{1}\leq n_{2}\leq \cdots n_{k}$ and $k\geq 0$ , whose levels are $N=\sum _{i=1}^{k}n_{i}$ . Any state whose level is not zero is called a descendant state of $v$ .

For any pair of complex numbers $h$ and $c$ , the Verma module ${\mathcal {V}}_{c,h}$ is the largest possible lowest 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 $L_{-n_{1}}L_{-n_{2}}\cdots L_{-n_{k}}v$ with $0<n_{1}\leq n_{2}\leq \cdots n_{k}$ and $k\geq 0$ 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 lowest weight representations with these values of $h$ and $c$ , called degenerate representations, which are cosets of the Verma module. In particular, the unique irreducible lowest weight representation with these values of $h$ and $c$ is the coset 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 lowest weight representation is a state that is both descendent and primary.

A sufficient condition for the Verma module ${\mathcal {V}}_{c,h}$ to have a singular vector at the level $N$ is $h=h_{r,s}(c)$ for some positive integers $r,s$ such that $N$ = $rs$ , with

h_{r,s}(c)={\frac {1}{4}}{\Big (}(b+b^{-1})^{2}-(br+b^{-1}s)^{2}{\Big )}\ ,\quad {\text{where}}\quad c=1+6(b+b^{-1})^{2}\ .

In particular, $h_{1,1}(c)=0$ , and the reducible Verma module ${\mathcal {V}}_{c,0}$ has a singular vector $L_{-1}v$ at the level $N$ =1. Then $h_{2,1}(c)=-{\frac {1}{2}}-{\frac {3}{4}}b^{2}$ , and the corresponding reducible Verma module has a singular vector $(L_{-1}^{2}+b^{2}L_{-2})v$ at the level $N$ =2.

Hermitian form and unitarity

A lowest weight representation with a real value of $c$ has a unique Hermitian form such that the adjoint of $L_{n}$ is $L_{{-n}}$ , 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 lowest weight representations are irreducible.

The Gram determinant of a basis of the level $N$ is given by the Kac determinant formula,

A_{N}\prod _{{1\leq r,s\leq N}}(h-h_{{r,s}}(c))^{{p(N-rs)}}~,

where the function p(N) is the partition function, and A_N is some constant. The Kac determinant formula was stated by V. Kac (1978), and its first published proof was given by Feigin and Fuks (1984).

The irreducible lowest weight representation with values $h$ and $c$ is unitary if and only if either $c$ ≥1 and $h$ ≥0, or $c$ is one of the values

c=1-{6 \over m(m+1)}=0,\quad 1/2,\quad 7/10,\quad 4/5,\quad 6/7,\quad 25/28,\ldots

for m = 2, 3, 4, ..., and h is one of the values

h=h_{{r,s}}(c)={((m+1)r-ms)^{2}-1 \over 4m(m+1)}

for r = 1, 2, 3, ..., m−1 and s= 1, 2, 3, ..., r.

Daniel Friedan, 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 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.

The unitary irreducible lowest weight representations with $c$ ≤ 1 are called the discrete series representations of the Virasoro algebra. These are special cases of the representations with m = q/(p−q), 0<r<q, 0< s<p for p and q coprime integers and r and s integers, called the minimal models and first studied in Belavin et al. (1984).

The first few discrete series representations are given by:

m = 2: c = 0, h = 0. The trivial representation.
m = 3: c = 1/2, h = 0, 1/16, 1/2. These 3 representations are related to the Ising model
m = 4: c = 7/10. h = 0, 3/80, 1/10, 7/16, 3/5, 3/2. These 6 representations are related to the tri critical Ising model.
m = 5: c = 4/5. There are 10 representations, which are related to the 3-state Potts model.
m = 6: c = 6/7. There are 15 representations, which are related to the tri critical 3-state Potts model.

Applications

Conformal field theory

In two dimensions, the algebra of local conformal transformations is made of two copies of the Witt algebra. It follows that the symmetry algebra of a quantum field theory with local conformal symmetry is the Virasoro algebra.

String theory

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. This is known as the Virasoro constraint, and in the quantum theory, cannot be applied to all the states in the theory, but rather only on the physical states (compare Gupta-Bleuler quantization).

Generalizations

There are two supersymmetric N=1 extensions of the Virasoro algebra, called the Neveu-Schwarz algebra and the Ramond algebra. Their theory is similar to that of the Virasoro algebra, now involving Grassmann numbers. There are further extensions of these algebras with more supersymmetry, such as the N = 2 superconformal algebra.

The Virasoro algebra is a central extension of the Lie algebra of meromorphic vector fields on a genus 0 Riemann surface that are holomorphic except at two fixed points. I V Krichever and S P Novikov (1987) found a central extension of the Lie algebra of meromorphic vector fields on a higher genus compact Riemann surface that are holomorphic except at two fixed points, and their analysis has been extended to supermanifolds by J Rabin (1995).

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. Unsurprisingly these are called the vertex Virasoro and conformal Virasoro algebras respectively.

The Virasoro algebra may also be specified as a presentation. This is to say that all of its generators may be determined recursively ("generated") out of merely two judiciously chosen generators (e.g. $L$ ₃ and $L$ ₋₂), and six equations (constraint conditions) among them, by systematic use of the Jacobi identity. (D Fairlie, J Nuyts, and C Zachos, 1988. Shortly thereafter, J Uretsky discovered the original 8 conditions could be pared down to six.)

Correspondingly, for the Super Virasoro algebra extension, the Ramond algebra follows from two generating generators and five conditions; and the Neveu-Schwarz algebra out of two such and nine conditions.

History

The Witt algebra (the Virasoro algebra without the central extension) was discovered by E. Cartan (1909). Its analogues over finite fields were studied by E. Witt in about the 1930s. The central extension of the Witt algebra that gives the Virasoro algebra was first found (in characteristic p>0) by R. E. Block (1966, page 381) and independently rediscovered (in characteristic 0) by I. M. Gelfand and D. B. Fuchs (1968). Virasoro (1970) wrote down some operators generating the Virasoso algebra while studying dual resonance models, though he did not find the central extension. The central extension giving the Virasoro algebra was rediscovered in physics shortly after by J. H. Weis, according to Brower and Thorn (1971, footnote on page 167).

References

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Virasoro algebra

Definition

Representation theory

Lowest weight representations