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 Lattice VOAs

Construction from an even positive-definite lattice.
Conformal structure and basic properties.

§7.3 Virasoro minimal models

Construction and basic properties.
Relation to suitable coset constructions.

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