Virasoro algebra

A highest weight representation of the Virasoro algebra is a representation generated by a primary state: a vector ${\displaystyle v}$ such that

${\displaystyle L_{n>0}v=0,\quad L_{0}v=hv,}$

where the number h is called the conformal dimension or conformal weight of ${\displaystyle v}$.[4]

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

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

For any pair of complex numbers h and c, the Verma module ${\displaystyle {\mathcal {V}}_{c,h}}$ 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 ${\displaystyle L_{-n_{1}}L_{-n_{2}}\cdots L_{-n_{k}}v}$ with ${\displaystyle 0<n_{1}\leq n_{2}\leq \cdots n_{k}}$ and ${\displaystyle k\geq 0}$ 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 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 descendant and primary.

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

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

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

This condition for the existence of a singular vector at the level ${\displaystyle N}$ is not necessary. In particular, there is a singular vector at the level ${\displaystyle N}$ if ${\displaystyle N=rs+r's'}$ with ${\displaystyle h=h_{r,s}(c)}$ and ${\displaystyle h+rs=h_{r',s'}(c)}$. This singular vector is now a descendant of another singular vector at the level ${\displaystyle rs}$. This type of singular vectors can however only exist if the central charge is of the type

${\displaystyle c=1-6{\frac {(p-q)^{2}}{pq}}\quad {\text{with}}\quad p,q\in \mathbb {Z} }$.

(For ${\displaystyle p>q\geq 2}$ coprime, these are the central charges of the minimal models.)[4]

A highest weight representation with a real value of ${\displaystyle c}$ has a unique Hermitian form such that the Hermitian adjoint of ${\displaystyle L_{n}}$ is ${\displaystyle L_{n}^{\dagger }=L_{-n}}$ 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 highest weight representations are irreducible.

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

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

where the function p(N) is the partition function, and ${\displaystyle A_{N}}$ is a positive constant that does not depend on ${\displaystyle h}$ or ${\displaystyle c}$. 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 h and c is unitary if and only if either c ≥ 1 and h ≥ 0, or

${\displaystyle c\in \left\{1-{\frac {6}{m(m+1)}}\right\}_{m=2,3,4,\ldots }=\left\{0,{\frac {1}{2}},{\frac {7}{10}},{\frac {4}{5}},{\frac {6}{7}},{\frac {25}{28}},\ldots \right\}}$

and h is one of the values

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

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 character of a representation ${\displaystyle {\mathcal {R}}}$ of the Virasoro algebra is the function

${\displaystyle \chi _{\mathcal {R}}(q)=\operatorname {Tr} _{\mathcal {R}}q^{L_{0}-{\frac {c}{24}}}.}$

The character of the Verma module ${\displaystyle {\mathcal {V}}_{c,h}}$ is

${\displaystyle \chi _{{\mathcal {V}}_{c,h}}(q)={\frac {q^{h-{\frac {c}{24}}}}{\prod _{n=1}^{\infty }(1-q^{n})}}={\frac {q^{h-{\frac {c-1}{24}}}}{\eta (q)}}=q^{h-{\frac {c}{24}}}\left(1+q+2q^{2}+3q^{3}+5q^{4}+\cdots \right),}$

where ${\displaystyle \eta }$ is the Dedekind eta function.

For any ${\displaystyle c\in \mathbb {C} }$ and for ${\displaystyle r,s\in \mathbb {N} ^{*}}$, the Verma module ${\displaystyle {\mathcal {V}}_{c,h_{r,s}}}$ is reducible due to the existence of a singular vector at level ${\displaystyle rs}$. This singular vector generates a submodule, which is isomorphic to the Verma module ${\displaystyle {\mathcal {V}}_{c,h_{r,s}+rs}}$. The quotient of ${\displaystyle {\mathcal {V}}_{c,h_{r,s}}}$ by this submodule is irreducible if ${\displaystyle {\mathcal {V}}_{c,h_{r,s}}}$ does not have other singular vectors, and its character is

${\displaystyle \chi _{{\mathcal {V}}_{c,h_{r,s}}/{\mathcal {V}}_{c,h_{r,s}+rs}}=\chi _{{\mathcal {V}}_{c,h_{r,s}}}-\chi _{{\mathcal {V}}_{c,h_{r,s}+rs}}=(1-q^{rs})\chi _{{\mathcal {V}}_{c,h_{r,s}}}.}$

Let ${\displaystyle c=c_{p,p'}}$ with ${\displaystyle 2\leq p<p'}$ and ${\displaystyle p,p'}$ coprime, and ${\displaystyle 1\leq r\leq p-1}$ and ${\displaystyle 1\leq s\leq p'-1}$. (Then ${\displaystyle (r,s)}$ is in the Kac table of the corresponding minimal model). The Verma module ${\displaystyle {\mathcal {V}}_{c,h_{r,s}}}$ has infinitely many singular vectors, and is therefore reducible with infinitely many submodules. 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

${\displaystyle {\begin{aligned}&\chi _{{\mathcal {V}}_{c,h_{r,s}}/({\mathcal {V}}_{c,h_{r,s}+rs}+{\mathcal {V}}_{c,h_{r,s}+(p-r)(p'-s)})}\\&=\sum _{k\in \mathbb {Z} }\left(\chi _{{\mathcal {V}}_{c,{\frac {1}{4pp'}}\left((p'r-ps+2kpp')^{2}-(p-p')^{2}\right)}}-\chi _{{\mathcal {V}}_{c,{\frac {1}{4pp'}}\left((p'r+ps+2kpp')^{2}-(p-p')^{2}\right)}}\right).\end{aligned}}}$

This expression is an infinite sum because the submodules ${\displaystyle {\mathcal {V}}_{c,h_{r,s}+rs}}$ and ${\displaystyle {\mathcal {V}}_{c,h_{r,s}+(p-r)(p'-s)}}$ have a nontrivial intersection, which is itself a complicated submodule.

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 two-dimensional conformal field theory is the Virasoro algebra. 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. 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 formalism).

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.

W-algebras are associative algebras which contain the Virasoro algebra, and which play an important role in two-dimensional conformal field theory. Among W-algebras, the Virasoro algebra has the particularity of being a Lie algebra.

The Virasoro algebra is a subalgebra of the universal enveloping algebra of any affine Lie algebra, as shown by the Sugawara construction. In this sense, affine Lie algebras are extensions of the Virasoro algebra.

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.[5] This can be further generalized to supermanifolds.[6]

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.