Introduction to Vertex Operator Algebras

--- Base version+++ Proposed version@@ -0,0 +1,374 @@+# 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. **Field supercommutator.**++   $$+   [Y(a,z),Y(b,w)]_{\mathrm s}+   =\sum_{j\geq0}\frac{Y(a_{(j)}b,w)}{j!}+   \partial_w^j\bigl(z^{-1}\delta(w/z)\bigr).+   $$++   Only finitely many terms contribute, by lower truncation.++4. **Zero-mode superderivations.**++   $$+   [a_{(0)},Y(b,z)]_{\mathrm s}=Y(a_{(0)}b,z),+   \qquad a_{(0)}\mathbf1=0.+   $$++   Equivalently, for $d\in V$,++   $$+   a_{(0)}\bigl(Y(b,z)d\bigr)+   =Y(a_{(0)}b,z)d+   +(-1)^{p(a)p(b)}Y(b,z)a_{(0)}d.+   $$++5. **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++$$+F=\Lambda(\xi_0,\xi_1,\xi_2,\ldots),\qquad \mathbf1=1,+$$++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/).++## 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 $B$.+:::