Introduction to Vertex Operator Algebras

--- Base version+++ Proposed version@@ -127,6 +127,143 @@ 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](https://archive.math.arizona.edu/cakeller/VertexOperatorAlgebras.pdf#page=25). ::: +## Fields of Virasoro descendants++::: proposition [Virasoro iterate formula] {@prop:virasoro-iterate}+For $a,b\in V$ and $n\in\mathbb Z$,++$$+\begin{aligned}+Y(L_na,z)b+=\operatorname{Res}_u\Bigl(&\iota_{u,z}(u-z)^{n+1}+Y(\omega,u)Y(a,z)b\\+&-\iota_{z,u}(u-z)^{n+1}+Y(a,z)Y(\omega,u)b\Bigr).+\end{aligned}+$$++The first kernel is expanded in nonnegative powers of $z$, and the second in nonnegative powers of $u$. The formula holds for arbitrary states; no primary-state assumption is needed.+:::++::: proof+Apply @prop:jacobi-identity to $\omega,a,b$, with field coordinates $u,z$ and relative coordinate $x$. Multiply by $x^{n+1}$ and take $\operatorname{Res}_x\operatorname{Res}_u$. For the iterate term, the $u$-residue of $u^{-1}\delta((z+x)/u)$ is $1$, so its remaining residue is++$$+\begin{aligned}+\operatorname{Res}_x x^{n+1}Y(Y(\omega,x)a,z)b+&=\operatorname{Res}_x x^{n+1}+\sum_{k\in\mathbb Z}Y(L_ka,z)b\,x^{-k-2}\\+&=Y(L_na,z)b.+\end{aligned}+$$++For the two product terms, the $x$-residues of the delta kernels are++$$+\begin{aligned}+\operatorname{Res}_x x^n\delta\!\left(\frac{u-z}{x}\right)+&=\iota_{u,z}(u-z)^{n+1},\\+\operatorname{Res}_x x^n\delta\!\left(\frac{z-u}{-x}\right)+&=\iota_{z,u}(u-z)^{n+1}.+\end{aligned}+$$++The remaining $u$-residue gives the stated identity.+:::++::: 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 $:A(:BC:):\ne:(:AB:)C:$.+:::++For the conformal field $\mathcal T(z):=Y(\omega,z)$, this convention gives++$$+\mathcal T_+(z)=\sum_{n\le-2}L_nz^{-n-2},\qquad+\mathcal T_-(z)=\sum_{n\ge-1}L_nz^{-n-2}.+$$++::: corollary [Normal-ordered descendant fields] {@cor:virasoro-descendants}+For $a\in V$ and $m\ge2$,++$$+Y(L_{-m}a,z)+=\frac1{(m-2)!}:\bigl(\partial_z^{m-2}\mathcal T(z)\bigr)Y(a,z):.+$$++For $m=1$, one has $Y(L_{-1}a,z)=\partial_zY(a,z)$. In particular,++$$+Y(L_{-2}a,z)=:\mathcal T(z)Y(a,z):,\qquad+Y(L_{-3}a,z)=:(\partial_z\mathcal T(z))Y(a,z):.+$$+:::++::: proof+Set $n=-m$ in @prop:virasoro-iterate and expand the kernels in the indicated orders. Taking the residues gives, for $b\in V$,++$$+\begin{aligned}+Y(L_{-m}a,z)b+&=\sum_{j\ge0}\binom{m+j-2}{j}z^j+L_{-m-j}Y(a,z)b\\+&\quad+(-1)^m\sum_{j\ge0}\binom{m+j-2}{j}+ z^{1-m-j}Y(a,z)L_{j-1}b.+\end{aligned}+$$++These are exactly the two terms in++$$+\frac1{(m-2)!}\Bigl(+(\partial_z^{m-2}\mathcal T_+(z))Y(a,z)++Y(a,z)(\partial_z^{m-2}\mathcal T_-(z))+\Bigr)b.+$$++Differentiation preserves the creation and annihilation split, so this is the claimed normal-ordered product. The first sum is well-defined coefficientwise by lower truncation of $Y(a,z)b$; the second is finite because $L_{j-1}b=0$ for sufficiently large $j$. The case $m=1$ is translation compatibility.+:::++With the right-nesting convention, repeated application of the corollary gives, for $m_1,\ldots,m_s\ge2$,++$$+Y(L_{-m_1}\cdots L_{-m_s}a,z)+=\frac{:\bigl(\partial_z^{m_1-2}\mathcal T(z)\bigr)\cdots+\bigl(\partial_z^{m_s-2}\mathcal T(z)\bigr)Y(a,z):}+{\prod_{i=1}^s(m_i-2)!}.+$$++For example,++$$+Y(L_{-2}^2a,z)=:\mathcal T(z)\mathcal T(z)Y(a,z):+=:\mathcal T(z)\bigl(:\mathcal T(z)Y(a,z):\bigr):.+$$++These identities are instances of the field-product theorem; see [Nozaradan, §4.3.3](https://arxiv.org/pdf/0809.1380).++The recursion removes one negative Virasoro mode from the state inserted at $z$. Applying it repeatedly treats $Y(L_{-m_1}\cdots L_{-m_s}a,z)b$. Inside a correlation function, the resulting Virasoro operators can be moved through other fields using @prop:virasoro-covariance and evaluated on the external states. With vacuum or highest-weight external states, this reduces descendant correlation functions to those of the underlying states. Those starting correlation functions remain input data.+ ## Weights and dilations  ::: proposition [Weight compatibility] {@prop:voa-weights}