Introduction to Vertex Operator Algebras

--- Base version+++ Proposed version@@ -150,30 +150,56 @@  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 $\sum_{j\geq0}Y(\omega_{(j)}\omega,w)/(z-w)^{j+1}$.-:::+## Products involving the conformal vector++::: proposition [Products of the conformal vector] {@prop:stress-tensor-ope}+The nonnegative products of $\omega$ with itself are++$$+\omega_{(0)}\omega=T\omega,\qquad+\omega_{(1)}\omega=2\omega,\qquad+\omega_{(2)}\omega=0,\qquad+\omega_{(3)}\omega=\frac c2\mathbf1,+$$++and $\omega_{(n)}\omega=0$ for $n\geq4$. Equivalently, the exact identity in $V((x))$ is++$$+Y(\omega,x)\omega+=\frac{c\mathbf1}{2x^4}+\frac{2\omega}{x^2}++\frac{T\omega}{x}++\sum_{k\geq0}L_{-k-2}\omega\,x^k.+$$+:::++::: proof+For $n\geq0$, use $\omega=L_{-2}\mathbf1$ and $L_n\mathbf1=0$ to obtain++$$+\begin{aligned}+L_n\omega+&=[L_n,L_{-2}]\mathbf1\\+&=(n+2)L_{n-2}\mathbf1++\frac{c}{12}(n^3-n)\delta_{n,2}\mathbf1.+\end{aligned}+$$++For $n=0,1,2$, this gives $2\omega,0,c\mathbf1/2$, respectively; for $n\geq3$ it vanishes. Together with $L_{-1}\omega=T\omega$ and $\omega_{(j)}=L_{j-1}$, these are the claimed products. Expanding $Y(\omega,x)\omega=\sum_nL_n\omega\,x^{-n-2}$ gives the full identity, including its regular terms.+:::++Applying the state-field map coefficientwise gives++$$+\begin{aligned}+Y(Y(\omega,x)\omega,w)+&=\frac{c\,\operatorname{id}_V}{2x^4}++\frac{2Y(\omega,w)}{x^2}++\frac{\partial_wY(\omega,w)}{x}\\+&\quad+\sum_{k\geq0}Y(L_{-k-2}\omega,w)x^k.+\end{aligned}+$$++After acting on any $b\in V$, @thm:iterate-expansion identifies this with the expansion at $x=0$ of the common expression for the two products of $Y(\omega,u)$ and $Y(\omega,w)$, with $u=w+x$. This is the precise expansion underlying the stress-tensor OPE.  ::: 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 $L_1a=0$.@@ -183,9 +209,9 @@ If $a\in V_h$ is primary, then  $$-\mathcal T(z)Y(a,w)\sim-\frac{hY(a,w)}{(z-w)^2}-+\frac{\partial_wY(a,w)}{z-w},+Y(\omega,x)a+=\frac{ha}{x^2}+\frac{Ta}{x}++\sum_{k\geq0}L_{-k-2}a\,x^k, $$  and, for every $m\in\mathbb Z$,@@ -199,10 +225,10 @@ :::  ::: 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 $c\ne0$.+In $Y(\omega,x)a=\sum_nL_na\,x^{-n-2}$, the primary-state conditions leave only $L_0a=ha$ and $L_{-1}a=Ta$ among the singular terms. The same conditions in @prop:virasoro-covariance give 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 since $L_1\omega=0$; it is not primary when $c\ne0$, since $L_2\omega=c\mathbf1/2$.  ## The weight-one Lie algebra