Chapter 3 of 7

§Vertex operator algebras

Although vertex algebra contains many familiar properties a chiral CFT constains, it does not contain the most important structure: the energy-momentum tensor and Virasoro algebra. A vertex operator algebra (VOA) is a vertex algebra with so-called conformal structure, which is essentially the energy-momentum tensor and Virasoro algebra. We now define VOA.

§3.1 Definition of VOA

Definition 3.1 (Vertex operator algebra)

Let $(V,Y,\mathbf1,T)$ be a vertex algebra. A vertex operator algebra (VOA) structure on $V$ is specified by a conformal vector $\omega\in V$ whose field

Y(\omega,z)=\sum_{n\in\mathbb Z}L_nz^{-n-2}, \qquad L_n=\omega_{(n+1)},

satisfies the following conditions.

Virasoro relations. For some $c\in\mathbb C$ $C$ , called the central charge,

$[L_m,L_n]=(m-n)L_{m+n} +\frac{c}{12}(m^3-m)\delta_{m+n,0}\operatorname{id}_V.$
Translation compatibility. $L_{-1}=T$ $T$ .
Conformal grading. $L_0$ acts diagonally with integral eigenvalues,

$V=\bigoplus_{h\in\mathbb Z}V_h, \qquad V_h=\{a\in V:L_0a=ha\},$

where $\dim V_h<\infty$ and $V_h=0$ for sufficiently negative $h$ $h$ . A state in $V_h$ has conformal weight $h$ $h$ .

We denote the VOA by $(V,Y,\mathbf1,\omega)$ $(V, Y, 1, ω)$ . The VOA is called CFT type if they satisfy additional conditions $V_h=0$ for $h<0$ and $V_0=\mathbb C\mathbf1$ $C 1$ .

§3.2 Vacuum and conformal vector

Proposition 3.2 (Vacuum and conformal weights)

The vacuum and conformal vector satisfy

L_n\mathbf1=0\quad(n\geq-1), \qquad \omega=L_{-2}\mathbf1, \qquad \mathbf1\in V_0,\quad\omega\in V_2.

Proof.

Apply Proposition 2.3 to $\omega$ and compare coefficients in $Y(\omega,z)\mathbf1=e^{zT}\omega$ $e^{z T} ω$ . Then $L_0\mathbf1=0$ and

L_0\omega=[L_0,L_{-2}]\mathbf1=2\omega.

§3.3 Virasoro action on fields

Proposition 3.3 (Virasoro covariance)

For $a\in V$ and $m\in\mathbb Z$ $Z$ ,

[L_m,Y(a,z)] =\sum_{j\geq0}\binom{m+1}{j}z^{m+1-j}Y(L_{j-1}a,z).

The sum is finite by lower truncation. In particular,

\begin{aligned} [L_{-1},Y(a,z)]&=\partial_zY(a,z),\\ [L_0,Y(a,z)]&=z\partial_zY(a,z)+Y(L_0a,z),\\ [L_1,Y(a,z)]&=z^2\partial_zY(a,z)+2zY(L_0a,z)+Y(L_1a,z). \end{aligned}

Proof.

Use Proposition 2.10 with first state $\omega$ and field coordinates $u,z$ $u, z$ . Taking $\operatorname{Res}_x$ turns each of the first two delta kernels into $1$ $1$ . Multiplying by $u^{m+1}$ and taking $\operatorname{Res}_u$ therefore gives

\operatorname{Res}_u u^{m+1}[Y(\omega,u),Y(a,z)] =[L_m,Y(a,z)].

On the right-hand side, the $u$ $u$ -residue of the delta kernel is

\operatorname{Res}_u u^{m+1}u^{-1} \delta\!\left(\frac{z+x}{u}\right) =\iota_{z,x}(z+x)^{m+1} =\sum_{j\geq0}\binom{m+1}{j}z^{m+1-j}x^j.

Thus the remaining $x$ $x$ -residue is

\begin{aligned} [L_m,Y(a,z)] &=\operatorname{Res}_x\left( \sum_{j\geq0}\binom{m+1}{j}z^{m+1-j}x^j\right) \left(\sum_{n\in\mathbb Z}Y(\omega_{(n)}a,z)x^{-n-1}\right)\\ &=\sum_{j\geq0}\binom{m+1}{j}z^{m+1-j}Y(\omega_{(j)}a,z). \end{aligned}

The residue selects $n=j$ $j$ . Since $\omega_{(j)}=L_{j-1}$ $L_{j - 1}$ , this is the claimed formula; lower truncation makes the sum finite even for $m<-1$ $- 1$ . The three special cases use $Y(L_{-1}a,z)=\partial_zY(a,z)$ $\partial_{z} Y (a, z)$ .

Remark 3.4 (Global conformal transformations)

The modes $L_{-1},L_0,L_1$ span an $\mathfrak{sl}_2$ subalgebra with no central term. Their actions on fields describe infinitesimal translations, dilations, and special conformal transformations, respectively. The corresponding coordinate flows are

z\longmapsto z+t,\qquad z\longmapsto e^t z, \qquad z\longmapsto\frac{z}{1-tz}.

Together they generate the Möbius transformations, the global holomorphic conformal transformations of the Riemann sphere. For $a\in V_h$ with $L_1a=0$ (a quasiprimary state), the three commutators act on $Y(a,z)$ as

\partial_z,\qquad z\partial_z+h,\qquad z^2\partial_z+2hz.

The terms involving $h$ account for the conformal weight of the field. For a general homogeneous state, the additional term $Y(L_1a,z)$ describes mixing with another field. See Keller, §7.3.

§3.4 Weights and dilations

Proposition 3.5 (Weight compatibility)

For $a\in V_h$ $V_{h}$ , $b\in V_k$ $V_{k}$ , and $n\in\mathbb Z$ $Z$ ,

[L_0,a_{(n)}]=(h-n-1)a_{(n)}, \qquad a_{(n)}b\in V_{h+k-n-1}.

In particular, $T V_h\subseteq V_{h+1}$ and $L_nV_h\subseteq V_{h-n}$ $V_{h - n}$ . For $q\in\mathbb C^\times$ $C^{\times}$ ,

q^{L_0}Y(a,z)q^{-L_0}=q^hY(a,qz).

Proof.

Compare coefficients in $[L_0,Y(a,z)]=(z\partial_z+h)Y(a,z)$ $h) Y (a, z)$ . Applying the resulting commutator to $b$ gives the weight of $a_{(n)}b$ $a_{(n)} b$ . The statements for $T=L_{-1}$ and $L_n$ follow from the Virasoro relations. Conjugation by $q^{L_0}$ multiplies $a_{(n)}$ by $q^{h-n-1}$ $q^{h - n - 1}$ , proving the last identity.

Thus $L_0$ generates dilations, and conformal weights determine the powers allowed in each OPE. If $V_\ell=0$ for $\ell<B$ $B$ , then $a_{(n)}b=0$ whenever $n>h+k-B-1$ $1$ .

§3.5 Stress tensor and primary fields

Write $\mathcal T(z)=Y(\omega,z)$ for the stress tensor, reserving $T$ for translation. Throughout this section, $\sim$ discards regular terms at $z=w$ $w$ .

Proposition 3.6 (Stress-tensor OPE)

\mathcal T(z)\mathcal T(w)\sim \frac{c\,\operatorname{id}_V}{2(z-w)^4} +\frac{2\mathcal T(w)}{(z-w)^2} +\frac{\partial_w\mathcal T(w)}{z-w}.

Proof.

Using $\omega=L_{-2}\mathbf1$ and the Virasoro relations gives

L_1\omega=0,\qquad L_2\omega=\frac c2\mathbf1, \qquad L_n\omega=0\quad(n\geq3).

Insert these, $L_0\omega=2\omega$ $2 ω$ , and $L_{-1}\omega=T\omega$ into the singular OPE $\sum_{j\geq0}Y(\omega_{(j)}\omega,w)/(z-w)^{j+1}$ $w)^{j + 1}$ .

Definition 3.7 (Primary and quasiprimary states)

A homogeneous state $a\in V_h$ is primary if $L_na=0$ for all $n>0$ $0$ , and quasiprimary if $L_1a=0$ $0$ .

Proposition 3.8 (Primary-field covariance)

If $a\in V_h$ is primary, then

\mathcal T(z)Y(a,w)\sim \frac{hY(a,w)}{(z-w)^2} +\frac{\partial_wY(a,w)}{z-w},

and, for every $m\in\mathbb Z$ $Z$ ,

[L_m,Y(a,z)] =\bigl(z^{m+1}\partial_z+(m+1)hz^m\bigr)Y(a,z).

For quasiprimary $a$ $a$ , the latter identity still holds for $m=-1,0,1$ $- 1, 0, 1$ .

Proof.

In the stress-tensor OPE, $\omega_{(j)}a=L_{j-1}a$ leaves only the translation and weight terms. The same substitution in Proposition 3.3 gives the commutator. For $m=-1,0,1$ $- 1, 0, 1$ , only $L_{-1}a,L_0a,L_1a$ can contribute.

The three modes $L_{-1},L_0,L_1$ describe infinitesimal translations, dilations, and special conformal transformations. The conformal vector itself is quasiprimary; the fourth-order pole shows that it is not primary when $c\ne0$ $0$ .

§3.6 The weight-one Lie algebra

Proposition 3.9 (Lie algebra and derivations)

If $V$ is of CFT type, then $V_1$ is a Lie algebra with bracket

[a,b]=a_{(0)}b.

Its zero modes act on $V$ by derivations:

[a_{(0)},Y(b,z)]=Y(a_{(0)}b,z), \qquad [a_{(0)},b_{(0)}]=(a_{(0)}b)_{(0)}.

The zero-mode action need not be faithful.

Proof.

Weight compatibility gives $a_{(0)}b\in V_1$ for $a,b\in V_1$ $V_{1}$ . Taking the $z^{-1}$ coefficient of skew-symmetry gives

a_{(0)}b=-b_{(0)}a+T(b_{(1)}a).

Here higher terms vanish by nonnegative weights, while $b_{(1)}a\in V_0=\mathbb C\mathbf1$ $C 1$ , so the last term is zero. Taking $\operatorname{Res}_x\operatorname{Res}_z$ in Proposition 2.10 gives the field commutator, and then taking its zero mode gives the second identity. Acting on $d\in V_1$ yields

[a,[b,d]]-[b,[a,d]]=[[a,b],d],

which is the Jacobi identity. The field commutator also proves the derivation property on all of $V$ $V$ .

§3.7 References

Christoph A. Keller, Introduction to Vertex Operator Algebras, §5.
Christophe Nozaradan, Introduction to Vertex Algebras, §3.7.

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