Chapter 7 of 11

§Examples of VOAs

§7.1 The Heisenberg VOA

The rank-one Heisenberg VOA $M(1)$ describes the current algebra of one chiral free boson. We construct its states and fields explicitly, then verify its conformal structure.

§The oscillator space

Definition 7.1 (Heisenberg vacuum space)

Let

M(1)=\mathbb C[x_1,x_2,\ldots],\qquad \mathbf1=1.

Every element is a polynomial in finitely many variables. Define operators, for $n>0$ $0$ , by

J_{-n}=x_n\cdot,\qquad J_n=n\frac{\partial}{\partial x_n},\qquad J_0=0.

They satisfy the Heisenberg relations

[J_m,J_n]=m\delta_{m+n,0}\operatorname{id},\qquad J_n\mathbf1=0\quad(n\geq0).

Equivalently, $M(1)$ has basis

J_{-n_1}\cdots J_{-n_s}\mathbf1, \qquad n_1\geq\cdots\geq n_s\geq1,

including the empty product $\mathbf1$ $1$ . Set $j=J_{-1}\mathbf1$ and

J(z)=\sum_{n\in\mathbb Z}J_nz^{-n-1}.

The polynomial realization specifies the action of every mode, not just its commutation relations. In particular, $J(z)v\in M(1)((z))$ for every $v\in M(1)$ $M (1)$ .

§The state–field correspondence

Proposition 7.2 (Heisenberg vertex algebra)

There is a unique vertex algebra structure on $M(1)$ with vacuum $\mathbf1$ and $Y(j,z)=J(z)$ $J (z)$ . Its fields are

Y(J_{-n_1}\cdots J_{-n_s}\mathbf1,z) =:\prod_{i=1}^s\frac{\partial_z^{n_i-1}J(z)}{(n_i-1)!}:, \qquad Y(\mathbf1,z)=\operatorname{id}.

The colons use the right-nested convention of Definition 2.12. For these factors, this amounts to moving all creation modes $J_{-n}$ to the left of all annihilation modes $J_n$ $J_{n}$ , $n>0$ $0$ . The translation operator is

T=\sum_{n\geq1}n x_{n+1}\frac{\partial}{\partial x_n}.

Proof.

The Heisenberg relations give

[J(z),J(w)]=\partial_w\bigl(z^{-1}\delta(w/z)\bigr)\operatorname{id}, \qquad (z-w)^2[J(z),J(w)]=0.

Also $T\mathbf1=0$ $0$ , $[T,J_n]=-nJ_{n-1}$ $- n J_{n - 1}$ , and $J(z)\mathbf1\in j+zM(1)[[z]]$ $z M (1) [[z]]$ . Thus $[T,J(z)]=\partial_zJ(z)$ $\partial_{z} J (z)$ . Derivatives and normal-ordered products preserve mutual locality by Dong’s lemma. The displayed fields consequently satisfy locality and translation covariance; their action on the vacuum is regular with constant term $J_{-n_1}\cdots J_{-n_s}\mathbf1$ $J_{- n_{1}} \dots J_{- n_{s}} 1$ . Lower truncation follows from the oscillator action. These are the vertex algebra axioms. Uniqueness follows from Corollary 2.13, which forces the displayed formula. See Keller, §3.2.

For example,

Y(J_{-2}\mathbf1,z)=\partial_zJ(z),\qquad Y(J_{-1}^2\mathbf1,z)=:J(z)^2:.

The complete product of the current with its state is

Y(j,z)j=z^{-2}\mathbf1+ \sum_{n\geq1}z^{n-1}J_{-n}J_{-1}\mathbf1.

Thus $j_{(1)}j=\mathbf1$ and $j_{(0)}j=0$ $0$ . This exact series specifies the regular terms as well as the pole.

§The conformal vector

Proposition 7.3 (Heisenberg conformal structure)

The vector

\omega=\frac12J_{-1}^2\mathbf1

makes $M(1)$ a VOA of CFT type and central charge $1$ $1$ . Its conformal field and modes are

Y(\omega,z)=\frac12:J(z)^2: =\sum_{m\in\mathbb Z}L_mz^{-m-2},\qquad L_m=\frac12\sum_{r\in\mathbb Z}:J_{m-r}J_r:.

Here mode normal ordering puts negative-index modes to the left. Each sum is finite on a fixed state. Moreover,

[L_m,J_n]=-nJ_{m+n},\qquad L_{-1}=T,\qquad L_0=\sum_{n>0}J_{-n}J_n.

In particular,

\operatorname{wt}(J_{-n_1}\cdots J_{-n_s}\mathbf1) =n_1+\cdots+n_s.

Proof.

Using the Heisenberg relations in the quadratic expression for $L_m$ gives

\begin{aligned} [L_m,J_n] &=\frac12\sum_r\left( r\delta_{r+n,0}J_{m-r} +(m-r)\delta_{m-r+n,0}J_r\right)\\ &=-nJ_{m+n}. \end{aligned}

The same quadratic expression yields

L_{-1}=\sum_{n>0}J_{-n-1}J_n=T,\qquad L_0=\sum_{n>0}n x_n\frac{\partial}{\partial x_n}.

It remains to check the Virasoro relations, including the central term. By the commutator just proved,

D_{m,n}:=[L_m,L_n]-(m-n)L_{m+n}

commutes with every $J_r$ $J_{r}$ . Any such operator is scalar: commuting with all derivatives forces its value on $\mathbf1$ to be constant, and commuting with all multiplications determines its value on every polynomial. Since $D_{m,n}$ has degree $-m-n$ for the grading above, it vanishes unless $m+n=0$ $0$ .

For $m>0$ $0$ , $L_m\mathbf1=L_0\mathbf1=0$ and

\begin{aligned} L_{-m}\mathbf1 &=\frac12\sum_{r=1}^{m-1}J_{-r}J_{-(m-r)}\mathbf1,\\ L_mL_{-m}\mathbf1 &=\frac12\sum_{r=1}^{m-1}r(m-r)\mathbf1 =\frac{m^3-m}{12}\mathbf1. \end{aligned}

Consequently,

[L_m,L_n]=(m-n)L_{m+n} +\frac{m^3-m}{12}\delta_{m+n,0}\operatorname{id}.

Finally, the weight- $h$ basis is indexed by partitions of $h$ $h$ , so every weight space is finite dimensional, negative weights vanish, and $M(1)_0=\mathbb C\mathbf1$ $C 1$ . All VOA axioms follow.

§Basic properties

Proposition 7.4 (Simplicity)

The VOA $M(1)$ is simple.

Proof.

An ideal is stable under every $J_n$ $J_{n}$ , hence under all polynomial multiplications and partial derivatives. Given a nonzero polynomial in the ideal, differentiate according to a monomial of maximal total degree. The result is a nonzero constant. Thus the ideal contains $\mathbf1$ and equals $M(1)$ $M (1)$ .

The first weight spaces are

\begin{aligned} M(1)_0&=\mathbb C\mathbf1,\\ M(1)_1&=\mathbb Cj,\\ M(1)_2&=\mathbb C J_{-2}\mathbf1\oplus\mathbb C J_{-1}^2\mathbf1 =\mathbb C Tj\oplus\mathbb C\omega. \end{aligned}

The weight-one Lie algebra is abelian with $\langle j,j\rangle=1$ $1$ , yet $j_{(0)}=J_0=0$ on all of $M(1)$ $M (1)$ . Thus its zero-mode action, and the associated inner automorphism group, are trivial even though its current algebra is nontrivial.

Exercise 7.5 (Reflection and composite products)

Show that $\theta(J_{-n_1}\cdots J_{-n_s}\mathbf1)=(-1)^sJ_{-n_1}\cdots J_{-n_s}\mathbf1$ is a VOA automorphism and that $\operatorname{Aut}(M(1))=\{1,\theta\}$ ${1, θ}$ . Identify the fixed-point space $M(1)^\theta$ in the oscillator basis.
For $u=J_{-1}^2\mathbf1$ $J_{- 1}^{2} 1$ , use $Y(u,z)=:J(z)^2:$ to verify

$u_{(-1)}u=J_{-1}^4\mathbf1+4J_{-3}J_{-1}\mathbf1.$

Thus a regular VOA product of composite states need not be their ordinary polynomial product.

§7.2 Virasoro minimal models

The Heisenberg VOA is generated by a weight-one current. We now construct a VOA generated by its conformal vector alone. At certain central charges, its vacuum descendants satisfy additional relations; imposing these relations gives the Virasoro minimal models.

§The universal vacuum VOA

Definition 7.6 (Virasoro vacuum representation)

Fix $c\in\mathbb C$ $C$ . Let $V^c$ be the representation of the Virasoro algebra generated by a vector $\mathbf1$ $1$ , subject only to the Virasoro relations with central charge $c$ and

L_n\mathbf1=0\qquad(n\geq-1).

By the Poincaré–Birkhoff–Witt theorem, a basis is

L_{-n_1}\cdots L_{-n_s}\mathbf1, \qquad n_1\geq\cdots\geq n_s\geq2,

including the empty product $\mathbf1$ $1$ . Set

\omega=L_{-2}\mathbf1,\qquad \mathcal T(z)=\sum_{n\in\mathbb Z}L_nz^{-n-2}.

The condition $L_{-1}\mathbf1=0$ is essential. A highest-weight vector of weight zero only satisfies $L_n\mathbf1=0$ for $n\geq0$ $0$ ; the vacuum additionally has zero translation. Thus $V^c$ is the weight-zero Verma module with the submodule generated by $L_{-1}\mathbf1$ removed.

Proposition 7.7 (Universal Virasoro VOA)

The space $V^c$ has a unique VOA structure with conformal vector $\omega$ and $Y(\omega,z)=\mathcal T(z)$ $T (z)$ . It is of CFT type, has central charge $c$ $c$ , and satisfies

\begin{aligned} Y(L_{-n_1}\cdots L_{-n_s}\mathbf1,z) &=:\prod_{i=1}^s \frac{\partial_z^{n_i-2}\mathcal T(z)}{(n_i-2)!}:,\\ \operatorname{wt}(L_{-n_1}\cdots L_{-n_s}\mathbf1) &=n_1+\cdots+n_s. \end{aligned}

The colons again mean right nesting. For any VOA $W$ of central charge $c$ $c$ , there is a unique VOA homomorphism

V^c\longrightarrow W,\qquad \mathbf1\longmapsto\mathbf1_W, \quad \omega\longmapsto\omega_W.

Its image is the vertex subalgebra generated by $\omega_W$ $ω_{W}$ .

Proof.

The Virasoro relations give $(z-w)^4[\mathcal T(z),\mathcal T(w)]=0$ and $[L_{-1},\mathcal T(z)]=\partial_z\mathcal T(z)$ $\partial_{z} T (z)$ . Also $\mathcal T(z)\mathbf1\in\omega+zV^c[[z]]$ $z V^{c} [[z]]$ , and the coefficients of iterated fields on $\mathbf1$ span $V^c$ $V^{c}$ . The reconstruction theorem therefore gives the vertex algebra with translation $T=L_{-1}$ $L_{- 1}$ . The normal-ordering formula follows from Corollary 2.13, using $L_{-n}=\omega_{(-n+1)}$ $ω_{(- n + 1)}$ . The Virasoro relations and the finite-dimensional PBW grading verify the conformal axioms. The map to $W$ sends each vacuum descendant to the same descendant of $\mathbf1_W$ $1_{W}$ ; the field formula shows that it preserves $Y$ $Y$ . See Keller, §§3.2 and 5.2.

§Singular vectors and the simple quotient

Definition 7.8 (Virasoro singular vector)

A nonzero homogeneous vector $\chi\in V^c$ of positive weight is singular if

L_n\chi=0\qquad(n>0).

Such a vector generates a proper Virasoro submodule: its descendants have weight at least $\operatorname{wt}(\chi)$ $wt (χ)$ , so they cannot contain the vacuum. Since $V^c$ is generated by $\omega$ $ω$ , this submodule is also a VOA ideal. In the quotient, the state $\chi$ and its entire field $Y(\chi,z)$ vanish.

The following representation-theoretic result supplies the central charges and the defining relation.

Theorem 7.9 (Virasoro vacuum singular-vector theorem)

The VOA $V^c$ is not simple precisely when

c=c_{p,q}:=1-\frac{6(p-q)^2}{pq}, \qquad 2\leq p<q,\quad \gcd(p,q)=1.

At this central charge, its unique maximal proper ideal is generated by a singular vector $\chi_{p,q}$ of weight

N_{p,q}=(p-1)(q-1).

This is the first positive weight at which $V^{c_{p,q}}$ has a singular vector, and $\chi_{p,q}$ is unique up to nonzero scalar.

We use this standard theorem without proof; see Ridout–Wood, §4. The general result uses the structure of Virasoro highest-weight representations. Once $p,q$ are fixed, the vector itself can be found by a finite calculation: write a linear combination of the PBW basis at weight $N_{p,q}$ and solve $L_1\chi=L_2\chi=0$ $0$ . These two equations imply $L_n\chi=0$ for every $n>0$ $0$ , since $L_1,L_2$ generate the positive Virasoro modes.

Definition 7.10 (Virasoro minimal model)

For coprime $2\leq p<q$ $q$ , the Virasoro minimal-model VOA is

\operatorname{Vir}_{p,q} :=V^{c_{p,q}}/(\chi_{p,q}),

where $(\chi_{p,q})$ is the VOA ideal generated by $\chi_{p,q}$ $χ_{p, q}$ . The quotient vacuum and conformal vector retain the notation $\mathbf1,\omega$ $1, ω$ . This simple VOA is also denoted $L(c_{p,q},0)$ $L (c_{p, q}, 0)$ .

Thus specifying $c_{p,q}$ in the Virasoro commutation relations constructs $V^{c_{p,q}}$ $V^{c_{p, q}}$ ; passing to the minimal model additionally imposes the null-field relation $Y(\chi_{p,q},z)=0$ $0$ .

§A defining relation at weight four

Example 7.11 (The Yang–Lee minimal model)

For $(p,q)=(2,5)$ $(2, 5)$ ,

c_{2,5}=-\frac{22}{5},\qquad N_{2,5}=4.

The weight-four space of $V^c$ has basis $L_{-2}^2\mathbf1,L_{-4}\mathbf1$ $L_{- 2}^{2} 1, L_{- 4} 1$ . Take

\chi=(L_{-2}^2+\alpha L_{-4})\mathbf1.

The Virasoro relations give

\begin{aligned} L_1\chi&=(3+5\alpha)L_{-3}\mathbf1,\\ L_2\chi&=(8+c+6\alpha)L_{-2}\mathbf1. \end{aligned}

The first equation forces $\alpha=-3/5$ $- 3/5$ ; the second then forces $c=-22/5$ $- 22/5$ . Hence

\chi_{2,5}=\left(L_{-2}^2-\frac35L_{-4}\right)\mathbf1, \qquad \operatorname{Vir}_{2,5}=V^{-22/5}/(\chi_{2,5}).

The state–field formula converts this relation into the exact field identity

:\mathcal T(z)^2:\;=\frac{3}{10}\partial_z^2\mathcal T(z).

Indeed, $Y(L_{-2}^2\mathbf1,z)=:\mathcal T(z)^2:$ and $Y(L_{-4}\mathbf1,z)=\tfrac12\partial_z^2\mathcal T(z)$ $\frac{1}{2} \partial_{z}^{2} T (z)$ . This equality holds in the quotient; its difference is a nonzero field in the universal VOA.

§Basic properties and further examples

Proposition 7.12 (Vacuum states of the minimal models)

Every $\operatorname{Vir}_{p,q}$ is simple, of CFT type, and generated by its conformal vector. The quotient map is an isomorphism on weight spaces below $N_{p,q}$ $N_{p, q}$ ; at weight $N_{p,q}$ its kernel is $\mathbb C\chi_{p,q}$ $C χ_{p, q}$ .

Except for $(p,q)=(2,3)$ $(2, 3)$ ,

(\operatorname{Vir}_{p,q})_1=0,\qquad (\operatorname{Vir}_{p,q})_2=\mathbb C\omega.

For $(p,q)=(2,3)$ $(2, 3)$ , $c=0$ $0$ , $\chi_{2,3}$ is proportional to $\omega$ $ω$ , and the quotient is the one-dimensional VOA $\mathbb C\mathbf1$ $C 1$ .

Proof.

Simplicity follows from Theorem 7.9. The ideal is spanned by negative-mode descendants of $\chi_{p,q}$ $χ_{p, q}$ , so it has no states below weight $N_{p,q}$ and has a one-dimensional component at that weight. Apply this to the PBW grading, whose first spaces are $V^c_0=\mathbb C\mathbf1$ $C 1$ , $V^c_1=0$ $0$ , and $V^c_2=\mathbb C\omega$ $C ω$ . If $N_{p,q}=2$ $2$ , the ideal contains the generator $\omega$ $ω$ , leaving only the vacuum.

Some small examples are:

$(p,q)$	Central charge	First vacuum null weight	Model
$(2,5)$	$-22/5$	$4$	Yang–Lee
$(3,4)$	$1/2$	$6$	Ising
$(4,5)$	$7/10$	$12$	Tricritical Ising

These VOAs have no weight-one currents. Their nontrivial conformal structure begins in weight two.

Exercise 7.13 (The Ising vacuum relation)

At $c=1/2$ $1/2$ , write a weight-six vector as

\chi=A L_{-2}^3\mathbf1+B L_{-3}^2\mathbf1 +C L_{-4}L_{-2}\mathbf1+D L_{-6}\mathbf1.

Solve $L_1\chi=L_2\chi=0$ for a nonzero vector, up to scale. Translate the resulting relation into an identity between right-nested normal-ordered products and derivatives of $\mathcal T(z)$ $T (z)$ .

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