Introduction to Vertex Operator Algebras

--- Base version+++ Proposed version@@ -37,3 +37,172 @@ We denote the VOA by $(V,Y,\mathbf1,\omega)$. The VOA is called **CFT type** if they satisfy additional conditions $V_h=0$ for $h<0$ and $V_0=\mathbb C\mathbf1$. ::: +## Vacuum and conformal vector++::: proposition [Vacuum and conformal weights] {@prop:voa-vacuum}+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 @prop:creation to $\omega$ and compare coefficients in $Y(\omega,z)\mathbf1=e^{zT}\omega$. Then $L_0\mathbf1=0$ and++$$+L_0\omega=[L_0,L_{-2}]\mathbf1=2\omega.+$$+:::++## Virasoro action on fields++::: proposition [Virasoro covariance] {@prop:virasoro-covariance}+For $a\in V$ and $m\in\mathbb Z$,++$$+[L_m,Y(a,z)]+=\sum_{j\geq0}\binom{m+1}{j}z^{m+1-j}Y(L_{j-1}a,z).+$$++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+Apply $\operatorname{Res}_x\operatorname{Res}_u u^{m+1}$ to @prop:jacobi-identity for the fields $Y(\omega,u)$ and $Y(a,z)$. Expanding $(z+x)^{m+1}$ in nonnegative powers of $x$ gives++$$+[\omega_{(m+1)},Y(a,z)]+=\sum_{j\geq0}\binom{m+1}{j}z^{m+1-j}Y(\omega_{(j)}a,z).+$$++Use $\omega_{(j)}=L_{j-1}$ and $Y(L_{-1}a,z)=\partial_zY(a,z)$.+:::++## Weights and dilations++::: proposition [Weight compatibility] {@prop:voa-weights}+For $a\in V_h$, $b\in V_k$, and $n\in\mathbb Z$,++$$+[L_0,a_{(n)}]=(h-n-1)a_{(n)},+\qquad a_{(n)}b\in V_{h+k-n-1}.+$$++In particular, $T V_h\subseteq V_{h+1}$ and $L_nV_h\subseteq V_{h-n}$. For $q\in\mathbb C^\times$,++$$+q^{L_0}Y(a,z)q^{-L_0}=q^hY(a,qz).+$$+:::++::: proof+Compare coefficients in $[L_0,Y(a,z)]=(z\partial_z+h)Y(a,z)$. Applying the resulting commutator to $b$ gives the weight of $a_{(n)}b$. The statements for $T=L_{-1}$ and $L_n$ follow from the Virasoro relations. Conjugation by $q^{L_0}$ multiplies $a_{(n)}$ by $q^{h-n-1}$, proving the last identity.+:::++Thus $L_0$ generates dilations, and conformal weights determine the powers allowed in each OPE. If $V_\ell=0$ for $\ell<B$, then $a_{(n)}b=0$ whenever $n>h+k-B-1$.++## 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 $z=w$.++::: proposition [Stress-tensor OPE] {@prop: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, $L_0\omega=2\omega$, and $L_{-1}\omega=T\omega$ into the singular OPE $\sum_{j\geq0}Y(\omega_{(j)}\omega,w)/(z-w)^{j+1}$.+:::++::: definition [Primary and quasiprimary states] {@def:primary-state}+A homogeneous state $a\in V_h$ is **primary** if $L_na=0$ for all $n>0$, and **quasiprimary** if $L_1a=0$.+:::++::: proposition [Primary-field covariance] {@prop:primary-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 $m\in\mathbb Z$,++$$+[L_m,Y(a,z)]+=\bigl(z^{m+1}\partial_z+(m+1)hz^m\bigr)Y(a,z).+$$++For quasiprimary $a$, the latter identity still holds for $m=-1,0,1$.+:::++::: proof+In the stress-tensor OPE, $\omega_{(j)}a=L_{j-1}a$ leaves only the translation and weight terms. The same substitution in @prop:virasoro-covariance gives the commutator. For $m=-1,0,1$, 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 $c\ne0$.++## The weight-one Lie algebra++::: proposition [Lie algebra and derivations] {@prop:voa-weight-one}+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 $a,b\in V_1$. 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 $b_{(1)}a\in V_0=\mathbb C\mathbf1$, so the last term is zero. Taking $\operatorname{Res}_x\operatorname{Res}_z$ in @prop:jacobi-identity 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 $V$.+:::++## References++- Christoph A. Keller, [Introduction to Vertex Operator Algebras](https://archive.math.arizona.edu/cakeller/VertexOperatorAlgebras.pdf), §5.+- Christophe Nozaradan, [Introduction to Vertex Algebras](https://arxiv.org/pdf/0809.1380), §3.7.