Introduction to Vertex Operator Algebras

--- Base version+++ Proposed version@@ -1,479 +0,0 @@-# Vertex operator superalgebras--Fermionic fields satisfy locality with a sign. A vertex operator superalgebra incorporates this sign through a parity grading. The free Majorana fermion provides the simplest example.--## Parity and locality--A **superspace** is a vector space $V=V_{\mathrm{even}}\oplus V_{\mathrm{odd}}$. A nonzero state $a$ is **parity-homogeneous** if it lies in one of these two subspaces; set $p(a)=0$ for an even state and $p(a)=1$ for an odd state. Here homogeneity refers to parity, not conformal weight. For parity-homogeneous operators $A,B$, the **supercommutator** is--$$-[A,B]_{\mathrm s}=AB-(-1)^{p(A)p(B)}BA.-$$--It is the usual commutator unless both operators are odd, in which case it is the anticommutator.--::: definition [Vertex superalgebra] {@def:vertex-superalgebra}-A **vertex superalgebra** (VSA) is a superspace $V$ with an even vacuum $\mathbf1$, an even translation operator $T$, and a linear state–field map--$$-Y(a,z)=\sum_{n\in\mathbb Z}a_{(n)}z^{-n-1}\qquad(a\in V),-$$--satisfying the truncation, vacuum, creation, and translation axioms of @def:vertex-algebra. For every parity-homogeneous $a$, each mode $a_{(n)}$ has parity $p(a)$: it preserves the two parity subspaces if $a$ is even and exchanges them if $a$ is odd. Locality is replaced by **super-locality**: for parity-homogeneous $a,b$, some $N\geq0$ satisfies--$$-(z-w)^N\left(Y(a,z)Y(b,w)--(-1)^{p(a)p(b)}Y(b,w)Y(a,z)\right)=0.-$$-:::--In particular, a product of two states of the same parity is even, and a product of two states of different parity is odd. Identities involving parity-homogeneous states extend to arbitrary states by linearity.--::: definition [Vertex operator superalgebra] {@def:vosa}-A **vertex operator superalgebra** (VOSA) is a vertex superalgebra with an even conformal vector $\omega$ such that--$$-Y(\omega,z)=\sum_{n\in\mathbb Z}L_nz^{-n-2}-$$--satisfies the Virasoro relations, $L_{-1}=T$, and $L_0$ is diagonalizable with a lower-bounded, finite-dimensional grading--$$-V=\bigoplus_{h\in\frac12\mathbb Z}V_h.-$$--We use the parity-compatible convention--$$-V_{\mathrm{even}}=\bigoplus_{h\in\mathbb Z}V_h,-\qquad-V_{\mathrm{odd}}=\bigoplus_{h\in\mathbb Z+\frac12}V_h.-$$--As before, CFT type means $V_h=0$ for $h<0$ and $V_0=\mathbb C\mathbf1$.-:::--The Virasoro algebra remains even. A VOSA does not by itself require an additional odd field generating an $N=1$ superconformal algebra. For these conventions, see [Gaudio, §2.1](https://inspirehep.net/files/0780521722085e7c87ff24bf4fa5e6f6#page=27).--## Normal ordering and basic properties--::: definition [Super normal ordering] {@def:super-normal-ordering}-Retain the split $A=A_++A_-$ from @def:normal-ordering, with $A_+$ containing the modes $A_{(n)}$ for $n<0$. For parity-homogeneous fields $A,B$, define--$$-:A(z)B(z):=A_+(z)B(z)-+(-1)^{p(A)p(B)}B(z)A_-(z).-$$--Multiple normal ordering means right nesting, with this sign at every step.-:::--The following properties parallel those of vertex algebras and VOAs. Their proofs are left as exercises. In every formula involving $p(a)$ or $p(b)$, the states $a,b$ are parity-homogeneous; extend the identities by linearity.--::: exercise [Properties of a vertex superalgebra] {@ex:vsa-properties}-Let $(V,Y,\mathbf1,T)$ be a VSA. Prove the following identities.--1. **Creation and translation.**--   $$-   Y(a,z)\mathbf1=e^{zT}a,\qquad-   Y(Ta,z)=\partial_zY(a,z).-   $$--   In particular, the state–field map is injective.--2. **Signed skew-symmetry.**--   $$-   Y(a,z)b=(-1)^{p(a)p(b)}e^{zT}Y(b,-z)a.-   $$--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.**--   $$-   [Y(a,z),Y(b,w)]_{\mathrm s}-   =\sum_{j\geq0}\frac{Y(a_{(j)}b,w)}{j!}-   \partial_w^j\bigl(z^{-1}\delta(w/z)\bigr).-   $$--   Obtain this by taking $\operatorname{Res}_x$ of the super Jacobi identity. Only finitely many terms contribute, by lower truncation.--5. **Zero-mode superderivations.**--   $$-   [a_{(0)},Y(b,z)]_{\mathrm s}=Y(a_{(0)}b,z),-   \qquad a_{(0)}\mathbf1=0.-   $$--   Equivalently, for $d\in V$,--   $$-   a_{(0)}\bigl(Y(b,z)d\bigr)-   =Y(a_{(0)}b,z)d-   +(-1)^{p(a)p(b)}Y(b,z)a_{(0)}d.-   $$--6. **Fields of negative products.** With super normal ordering,--   $$-   Y(a_{(-k-1)}b,z)=\frac1{k!}:-   \bigl(\partial_z^kY(a,z)\bigr)Y(b,z):,-   \qquad k\geq0.-   $$--   Deduce the corresponding formula for repeated negative products using right nesting.-:::--::: exercise [Properties of a vertex operator superalgebra] {@ex:vosa-properties}-Let $(V,Y,\mathbf1,\omega)$ be a VOSA with the parity-compatible grading of @def:vosa. Prove the following properties.--1. **Vacuum and conformal weights.**--   $$-   L_n\mathbf1=0\quad(n\geq-1),\qquad-   \omega=L_{-2}\mathbf1\in V_2.-   $$--   For $a\in V_h$ and $b\in V_k$,--   $$-   a_{(n)}b\in V_{h+k-n-1},\qquad TV_h\subseteq V_{h+1}.-   $$--2. **Virasoro covariance.** For every $a\in V$ and $m\in\mathbb Z$,--   $$-   [L_m,Y(a,z)]-   =\sum_{j\geq0}\binom{m+1}{j}z^{m+1-j}Y(L_{j-1}a,z).-   $$--   No parity sign appears because $\omega$ is even. In particular, for a primary state $a$ of weight $h$,--   $$-   [L_m,Y(a,z)]-   =\bigl(z^{m+1}\partial_z+(m+1)hz^m\bigr)Y(a,z).-   $$--3. **Parity and the even VOA.** The linear map $\sigma$, equal to $+1$ on $V_{\mathrm{even}}$ and $-1$ on $V_{\mathrm{odd}}$, is a VOSA automorphism. Its fixed-point space $V_{\mathrm{even}}$, with the restricted fields and the same vacuum and conformal vector, is an ordinary VOA of the same central charge. Moreover,--   $$-   \sigma=e^{2\pi iL_0}.-   $$--4. **Weight-one-half fermions.** Assume in addition that $V$ is of CFT type. For $a,b\in V_{\frac12}$, define $B(a,b)\in\mathbb C$ by--   $$-   a_{(0)}b=B(a,b)\mathbf1.-   $$--   Show that $B$ is a symmetric bilinear form, possibly degenerate, and--   $$-   a_{(j)}b=0\quad(j\geq1),\qquad-   \{Y(a,z),Y(b,w)\}-   =B(a,b)z^{-1}\delta(w/z)\operatorname{id}_V.-   $$--   Thus weight-one-half fields satisfy Clifford relations. Also show that every state in $V_{\frac12}$ is primary.-:::--## The free Majorana fermion--### Clifford modes and the vacuum space--::: definition [Fermionic vacuum space] {@def:majorana-vacuum}-Let--$$-F=\bigwedge(\xi_0,\xi_1,\xi_2,\ldots),\qquad \mathbf1=1,-$$--where all generators $\xi_n$ are odd and each state is a finite linear combination of exterior monomials. Define odd operators, for $n\geq0$, by--$$-\psi_{-n-\frac12}=\xi_n\wedge,-\qquad-\psi_{n+\frac12}=\frac{\partial}{\partial\xi_n}.-$$--:::--The derivative is the left exterior derivative: removing the $i$th factor of a monomial contributes the sign $(-1)^{i-1}$. Thus--$$-\{\psi_r,\psi_s\}=\delta_{r+s,0}\operatorname{id},-\qquad r,s\in\mathbb Z+\frac12,-\qquad \psi_r\mathbf1=0\quad(r>0).-$$--Set $\psi=\psi_{-\frac12}\mathbf1$ and--$$-\psi(z)=\sum_{r\in\mathbb Z+\frac12}\psi_rz^{-r-\frac12}.-$$--The half-integer labels are the usual Neveu–Schwarz mode convention. The field still has integer powers of $z$: in vertex-algebra notation, $\psi_{(n)}=\psi_{n+\frac12}$. Its creation part consists of $r<0$ and its annihilation part of $r>0$.--A basis of $F$ is--$$-\psi_{-n_1-\frac12}\cdots\psi_{-n_s-\frac12}\mathbf1,-\qquad n_1>\cdots>n_s\geq0.-$$--Its parity is $s\bmod2$. The strict inequalities express the fact that a fermionic creation mode cannot occur twice.--### Fields and conformal structure--::: proposition [Majorana vertex superalgebra] {@prop:majorana-superalgebra}-There is a unique vertex superalgebra structure on $F$ with $Y(\psi,z)=\psi(z)$. Its fields are--$$-Y(\psi_{-n_1-\frac12}\cdots\psi_{-n_s-\frac12}\mathbf1,z)-=:\prod_{i=1}^s\frac{\partial_z^{n_i}\psi(z)}{n_i!}:,-\qquad Y(\mathbf1,z)=\operatorname{id},-$$--using signed right-nested normal ordering. Translation is the even derivation determined by--$$-T\mathbf1=0,\qquad T\xi_n=(n+1)\xi_{n+1}.-$$-:::--::: proof-The Clifford relations give the formal anticommutator--$$-\{\psi(z),\psi(w)\}=z^{-1}\delta(w/z)\operatorname{id},-\qquad (z-w)\{\psi(z),\psi(w)\}=0.-$$--The oscillator action also gives $[T,\psi(z)]=\partial_z\psi(z)$ and $\psi(z)\mathbf1\in\psi+zF[[z]]$. The superalgebra reconstruction theorem applies, just as in @prop:heisenberg-va, with the Clifford relations supplying super-locality. The negative-product formula determines the fields of all basis states.-:::--For example, the full fermion-state product is--$$-Y(\psi,z)\psi=\frac{\mathbf1}{z}-+\sum_{n\geq1}z^n\psi_{-n-\frac12}\psi_{-\frac12}\mathbf1.-$$--In particular, $\psi_{(0)}\psi=\mathbf1$ but $\psi_{(-1)}\psi=0$, so $:\psi(z)\psi(z):=0$.--::: proposition [Majorana conformal vector] {@prop:majorana-conformal}-The even vector--$$-\omega=\frac12\psi_{-\frac32}\psi_{-\frac12}\mathbf1-$$--makes $F$ a VOSA of CFT type with central charge $c=\tfrac12$. Its conformal field is--$$-Y(\omega,z)=\frac12:(\partial_z\psi(z))\psi(z):-=\sum_{m\in\mathbb Z}L_mz^{-m-2}.-$$--The fermion is primary of weight $\tfrac12$, and--$$-[L_m,\psi_r]=-\left(r+\frac m2\right)\psi_{m+r},-\qquad-\operatorname{wt}(\psi_{-n_1-\frac12}\cdots\psi_{-n_s-\frac12}\mathbf1)-=\sum_{i=1}^s\left(n_i+\frac12\right).-$$-:::--::: proof-Expanding the conformal field and using the Clifford relations gives--$$-L_m=\frac12\sum_{r\in\mathbb Z+\frac12}-\left(r-\frac m2\right):\psi_{m-r}\psi_r:.-$$--Mode normal ordering moves annihilation modes to the right with a minus sign when two odd modes are exchanged. The sum is finite on each state. To compute its commutator with $\psi_s$, use--$$-[\psi_a\psi_b,\psi_s]-=\psi_a\{\psi_b,\psi_s\}-\{\psi_a,\psi_s\}\psi_b.-$$--The two Kronecker deltas yield $[L_m,\psi_s]=-(s+m/2)\psi_{m+s}$. Hence $L_{-1}=T$, since both kill the vacuum and have the same commutators with every creation mode. Also--$$-L_0=\sum_{r>0}r\psi_{-r}\psi_r,-$$--which gives the stated grading and parity. Each weight has finitely many basis states.--The commutator further gives--$$-L_0\omega=2\omega,\quad L_1\omega=0,\quad-L_2\omega=\frac14\mathbf1,\quad L_n\omega=0\ (n\geq3).-$$--For the central coefficient, explicitly,--$$-L_2\omega=\frac12\left(-\frac12\psi_{\frac12}\psi_{-\frac12}--\frac12\psi_{-\frac32}\psi_{\frac32}\right)\mathbf1-=\frac14\mathbf1.-$$--Since $\omega$ is even, the ordinary Jacobi commutator formula applies to its field. The products above, together with $L_{-1}\omega=T\omega$, give--$$-[L_m,L_n]=(m-n)L_{m+n}-+\frac{m^3-m}{24}\delta_{m+n,0}\operatorname{id}.-$$--Thus $c=\tfrac12$. Finally $L_0\psi=\tfrac12\psi$ and $L_n\psi=0$ for $n>0$, proving primarity.-:::--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}-The VOSA $F$ is simple.-:::--::: proof-An ideal is stable under all creation and annihilation modes. From a nonzero exterior polynomial, choose a monomial of maximal exterior degree and apply the annihilation modes corresponding to its factors. This produces a nonzero multiple of $\mathbf1$. Creation modes then generate all of $F$.-:::--The first weight spaces are--$$-F_0=\mathbb C\mathbf1,\quad F_{\frac12}=\mathbb C\psi,-\quad F_1=0,\quad F_{\frac32}=\mathbb C T\psi,-\quad F_2=\mathbb C\omega.-$$--Its even part has a familiar identification:--::: theorem [The even fermion algebra] {@thm:majorana-ising}-The even subalgebra of the Majorana VOSA is the Ising VOA:--$$-F_{\mathrm{even}}=\operatorname{Vir}(\omega)-\cong L\!\left(\frac12,0\right)-=\operatorname{Vir}_{3,4}.-$$-:::--This is the standard free-fermion realization of the Ising VOA; we quote the identification rather than infer it merely from the central charge. See [Adamović–Möller, §4.2](https://link.springer.com/article/10.1007/s00209-026-04078-5). The full $F$ also contains the odd field $\psi$ of weight $\tfrac12$, so it is a VOSA extension of the Ising VOA, not the same algebra.--::: exercise [Parity and fermionic products] {@ex:majorana-products}-1. Verify directly that $\sigma(\psi)=-\psi$ preserves the Clifford relations and fixes $\omega$. Identify $F_{\mathrm{even}}$ in the oscillator basis.-2. Compute $\psi_{(-2)}\psi$ and verify $:(\partial_z\psi(z))\psi(z):=2Y(\omega,z)$.-3. Apply @ex:vosa-properties to $F_{\frac12}=\mathbb C\psi$ and determine the form $B$.-:::