Introduction to Vertex Operator Algebras

--- Base version+++ Proposed version@@ -362,3 +362,121 @@  by $P(z,w)/(x^Nz^rw^s)$. This series has a uniform lower bound in each variable, so the products are well-defined. Delta substitution turns the first two terms into the product expansions of $F$ and the third into its iterate expansion, with exactly the conventions stated above. This proves the identity. :::++## Fields of products++::: proposition [Iterate formula] {@prop:field-iterate}+For $a,b,c\in V$ and $m\in\mathbb Z$,++$$+\begin{aligned}+Y(a_{(m)}b,z)c+=\operatorname{Res}_u\Bigl(&\iota_{u,z}(u-z)^m+Y(a,u)Y(b,z)c\\+&-\iota_{z,u}(u-z)^m+Y(b,z)Y(a,u)c\Bigr).+\end{aligned}+$$++The first kernel is expanded in nonnegative powers of $z$, and the second in nonnegative powers of $u$.+:::++::: proof+Apply @prop:jacobi-identity with field coordinates $u,z$ and relative coordinate $x$. Multiply by $x^m$ and take $\operatorname{Res}_x\operatorname{Res}_u$. For the iterate term, the $u$-residue of $u^{-1}\delta((z+x)/u)$ is $1$, leaving++$$+\begin{aligned}+\operatorname{Res}_x x^mY(Y(a,x)b,z)c+&=\operatorname{Res}_x x^m+\sum_{j\in\mathbb Z}Y(a_{(j)}b,z)c\,x^{-j-1}\\+&=Y(a_{(m)}b,z)c.+\end{aligned}+$$++For the two product terms,++$$+\begin{aligned}+\operatorname{Res}_x x^{m-1}\delta\!\left(\frac{u-z}{x}\right)+&=\iota_{u,z}(u-z)^m,\\+\operatorname{Res}_x x^{m-1}\delta\!\left(\frac{z-u}{-x}\right)+&=\iota_{z,u}(u-z)^m.+\end{aligned}+$$++Taking the remaining $u$-residue proves the formula.+:::++For $m\ge0$, the kernel is a polynomial, so its two expansions agree and the formula simplifies to++$$+Y(a_{(m)}b,z)+=\operatorname{Res}_u (u-z)^m[Y(a,u),Y(b,z)].+$$++These are the singular OPE coefficients. Negative products are expressed by normal ordering as follows.++::: definition [Normal ordering] {@def:normal-ordering}+For a field $A(z)=\sum_j A_{(j)}z^{-j-1}$, write++$$+A_+(z)=\sum_{j<0}A_{(j)}z^{-j-1},\qquad+A_-(z)=\sum_{j\ge0}A_{(j)}z^{-j-1}.+$$++Here $A_+$ is the creation part and $A_-$ the annihilation part. The **normal-ordered product** with a field $B(z)$ is++$$+:A(z)B(z):=A_+(z)B(z)+B(z)A_-(z).+$$++For several fields, **multiple normal ordering** means right nesting:++$$+:A_1(z)\cdots A_k(z):+:=:A_1(z)\bigl(:A_2(z)\cdots A_k(z):\bigr):,+\qquad :A_1(z):=A_1(z).+$$++Thus $:ABC:=:A(:BC:):$. This is a choice of parentheses, not an associativity identity; in general $:A(:BC:):\ne:(:AB:)C:$.+:::++::: corollary [Fields of negative products] {@cor:normal-ordered-products}+For $a,b\in V$ and $k\ge0$,++$$+Y(a_{(-k-1)}b,z)+=\frac1{k!}:\bigl(\partial_z^kY(a,z)\bigr)Y(b,z):.+$$++In particular, $Y(a_{(-1)}b,z)=:Y(a,z)Y(b,z):$.+:::++::: proof+For $m=-1$, the kernels in @prop:field-iterate are++$$+\iota_{u,z}(u-z)^{-1}=\sum_{j\ge0}z^ju^{-j-1},\qquad+\iota_{z,u}(u-z)^{-1}=-\sum_{j\ge0}u^jz^{-j-1}.+$$++Their residues give, respectively, the creation part of $Y(a,z)$ on the left and the annihilation part on the right:++$$+Y(a_{(-1)}b,z)+=Y(a,z)_+Y(b,z)+Y(b,z)Y(a,z)_-.+$$++Translation gives $Y(T^ka,z)=\partial_z^kY(a,z)$ and, on comparing the coefficient of $z^0$, $(T^ka)_{(-1)}=k!a_{(-k-1)}$. Apply the case $k=0$ to $T^ka,b$ and divide by $k!$.+:::++With the right-nesting convention, repeated application gives, for $k_1,\ldots,k_s\ge0$,++$$+Y\bigl(a^1_{(-k_1-1)}\cdots a^s_{(-k_s-1)}b,z\bigr)+=\frac{:\bigl(\partial_z^{k_1}Y(a^1,z)\bigr)\cdots+\bigl(\partial_z^{k_s}Y(a^s,z)\bigr)Y(b,z):}+{k_1!\cdots k_s!}.+$$++Thus fields of states built from negative products are nested normal-ordered derivatives of the original fields. The general iterate formula also handles nonnegative products through their singular OPE coefficients. These identities hold in any vertex algebra, without a conformal vector; see [Nozaradan, §4.3.3](https://arxiv.org/pdf/0809.1380).