Introduction to Vertex Operator Algebras

--- Base version+++ Proposed version@@ -88,7 +88,19 @@    Y(a,z)b=(-1)^{p(a)p(b)}e^{zT}Y(b,-z)a.    $$ -3. **Field supercommutator.**+3. **Super Jacobi identity.** For parity-homogeneous $a,b$ and any $c\in V$,++   $$+   \begin{aligned}+   &x^{-1}\delta\!\left(\frac{z-w}{x}\right)Y(a,z)Y(b,w)c\\+   &\quad-(-1)^{p(a)p(b)}x^{-1}\delta\!\left(\frac{w-z}{-x}\right)Y(b,w)Y(a,z)c\\+   &\qquad=z^{-1}\delta\!\left(\frac{w+x}{z}\right)Y(Y(a,x)b,w)c.+   \end{aligned}+   $$++   The binomials in the three delta kernels are expanded in nonnegative powers of $w$, $z$, and $x$, respectively, as in @prop:jacobi-identity. Only the reversed product acquires a parity sign; no homogeneity assumption on $c$ is needed. Adapt the proof of @prop:jacobi-identity using super-locality.++4. **Field supercommutator.**     $$    [Y(a,z),Y(b,w)]_{\mathrm s}@@ -96,9 +108,9 @@    \partial_w^j\bigl(z^{-1}\delta(w/z)\bigr).    $$ -   Only finitely many terms contribute, by lower truncation.--4. **Zero-mode superderivations.**+   Obtain this by taking $\operatorname{Res}_x$ of the super Jacobi identity. Only finitely many terms contribute, by lower truncation.++5. **Zero-mode superderivations.**     $$    [a_{(0)},Y(b,z)]_{\mathrm s}=Y(a_{(0)}b,z),@@ -113,7 +125,7 @@    +(-1)^{p(a)p(b)}Y(b,z)a_{(0)}d.    $$ -5. **Fields of negative products.** With super normal ordering,+6. **Fields of negative products.** With super normal ordering,     $$    Y(a_{(-k-1)}b,z)=\frac1{k!}:@@ -335,6 +347,98 @@  This is the **free Majorana fermion VOSA**, also called the neutral free fermion. The construction is over $\mathbb C$, as in the rest of the note. See [Gaudio, Example 6.1.5](https://inspirehep.net/files/0780521722085e7c87ff24bf4fa5e6f6#page=83) and [Bouchard et al., Eq. (4.68)](https://pmc.ncbi.nlm.nih.gov/articles/PMC7584568/). +## Wick's theorem and Pfaffian correlation functions++Let $\varepsilon:F\to\mathbb C$ extract the vacuum coefficient: $\varepsilon(\mathbf1)=1$ and $\varepsilon$ vanishes on all positive-weight states. Define the vacuum expectation by++$$+\langle A_1(z_1)\cdots A_n(z_n)\rangle+:=\varepsilon\bigl(A_1(z_1)\cdots A_n(z_n)\mathbf1\bigr).+$$++We display these correlators as rational functions; the ordered field product means their expansion in $|z_1|>\cdots>|z_n|$. In particular,++$$+\langle\psi(z)\psi(w)\rangle=\frac1{z-w}.+$$++::: proposition [Fermionic Wick theorem] {@prop:majorana-wick}+All odd vacuum correlation functions of $\psi$ vanish. For $2m$ insertions,++$$+\langle\psi(z_1)\cdots\psi(z_{2m})\rangle+=\operatorname{Pf}(A),\qquad+A_{ij}=\begin{cases}+\displaystyle\frac1{z_i-z_j},&i\ne j,\\+0,&i=j.+\end{cases}+$$++Here $A$ is antisymmetric and its **Pfaffian** is++$$+\operatorname{Pf}(A)+=\frac1{2^m m!}\sum_{\pi\in S_{2m}}\operatorname{sgn}(\pi)+\prod_{k=1}^m A_{\pi(2k-1),\pi(2k)}.+$$++Equivalently, sum over all pairings of the insertions, multiply the two-point functions of the pairs, and attach the sign of the permutation that brings the paired fermions together. The empty correlation function is $1$.+:::++::: proof+With the creation/annihilation convention already fixed, $\varepsilon\psi_+(z)=0$ and $\psi_-(z)\mathbf1=0$. The Clifford relations give++$$+\{\psi_-(z_1),\psi(z_j)\}+=\sum_{n\geq0}z_1^{-n-1}z_j^n\operatorname{id}+=\iota_{z_1,z_j}\frac1{z_1-z_j}\operatorname{id}.+$$++Write $G_n(z_1,\ldots,z_n)=\langle\psi(z_1)\cdots\psi(z_n)\rangle$. Only the annihilation part of the first field contributes. Moving it to the right gives one contraction with each later field; the final uncontracted term kills the vacuum. Thus, in rational-function notation,++$$+G_n(z_1,\ldots,z_n)+=\sum_{j=2}^{n}\frac{(-1)^j}{z_1-z_j}+G_{n-2}(z_2,\ldots,\widehat{z_j},\ldots,z_n).+$$++The hat means omission. The sign counts the $j-2$ intervening odd fields. Starting with $G_0=1$ and $G_1=0$, this recurrence makes every odd correlator zero and is exactly the expansion of the Pfaffian along its first row.+:::++For example, the four-point function is++$$+\begin{aligned}+\langle\psi(z_1)\psi(z_2)\psi(z_3)\psi(z_4)\rangle+&=\frac1{(z_1-z_2)(z_3-z_4)}\\+&\quad-\frac1{(z_1-z_3)(z_2-z_4)}++\frac1{(z_1-z_4)(z_2-z_3)}.+\end{aligned}+$$++This is the Pfaffian form of the chiral Majorana correlator; see [Ardonne–Sierra, §2](https://staff.fysik.su.se/~ardonne/files/papers/JPhysA_43_505402_2010.pdf).++Every field of $F$ is a linear combination of normal-ordered derivatives of $\psi$. Its vacuum correlations are therefore also computable by Wick pairings. The differentiated contraction is++$$+\langle\partial_z^a\psi(z)\,\partial_w^b\psi(w)\rangle+=\partial_z^a\partial_w^b\frac1{z-w}+=\frac{(-1)^a(a+b)!}{(z-w)^{a+b+1}},\qquad a,b\geq0.+$$++For products of normal-ordered fields, sum over complete pairings of their constituent fermions, allowing contractions only between different normal-ordered factors. The signs are those of the ordered constituent fermions. This computes correlations of arbitrary states in $F$, including the conformal vector and its descendants.++::: exercise [Stress-tensor two-point function] {@ex:majorana-stress-correlator}+Use $\mathcal T(z)=Y(\omega,z)=\tfrac12:(\partial_z\psi(z))\psi(z):$ and the differentiated contractions to show++$$+\langle\mathcal T(z)\mathcal T(w)\rangle+=\frac1{4(z-w)^4}.+$$++Identify the two allowed pairings and their signs. Compare the coefficient with $c/2$ for $c=\tfrac12$.+:::+ ## Simplicity and the Ising VOA  ::: proposition [Simplicity of the free fermion] {@prop:majorana-simple}