Introduction to Vertex Operator Algebras

--- Base version+++ Proposed version@@ -127,143 +127,6 @@ 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}