Vertex Operator Superalgebras and the Majorana Fermion

--- Base version+++ Proposed version@@ -0,0 +1,223 @@+# Definitions and properties++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. We work over $\mathbb C$ and assume familiarity with formal calculus. The formal delta function is $\delta(x)=\sum_{m\in\mathbb Z}x^m$.++## 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, for $a,b\in V$,++$$+\begin{gathered}+Y(a,z)b\in V((z)),\qquad+Y(\mathbf1,z)=\operatorname{id}_V,\qquad+Y(a,z)\mathbf1\in a+zV[[z]],\\+T\mathbf1=0,\qquad [T,Y(a,z)]=\partial_zY(a,z).+\end{gathered}+$$++The vacuum is nonzero. 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. Finally, **super-locality** requires that, 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++$$+[L_m,L_n]=(m-n)L_{m+n}++\frac{c}{12}(m^3-m)\delta_{m+n,0}\operatorname{id}_V+$$++for a scalar $c$, called the **central charge**, and $L_{-1}=T$. The operator $L_0$ is diagonalizable with a lower-bounded grading by finite-dimensional eigenspaces++$$+V=\bigoplus_{h\in\frac12\mathbb Z}V_h.+$$++We use the parity-compatible convention++$$+V_{\mathrm{even}}=\bigoplus_{h\in\mathbb Z}V_h,+\qquad+V_{\mathrm{odd}}=\bigoplus_{h\in\mathbb Z+\frac12}V_h.+$$++**CFT type** means $V_h=0$ for $h<0$ and $V_0=\mathbb C\mathbf1$.+:::++An ordinary VOA is the purely even case, with integral conformal weights. A state $a\in V_h$ is **primary** if $L_ma=0$ for all $m>0$. A VOSA automorphism is an even linear bijection $g$ fixing $\mathbf1$ and $\omega$ and satisfying $gY(a,z)b=Y(ga,z)gb$.++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).++An **ideal** of a VOSA $V$ is a parity-graded subspace $I$ satisfying $Y(a,z)I\subseteq I((z))$ for every $a\in V$. In particular, it is stable under $T=L_{-1}$. The VOSA is **simple** if it has no nonzero proper ideal.++## Normal ordering and basic properties++::: definition [Super normal ordering] {@def:super-normal-ordering}+For a field $A(z)=\sum_{m\in\mathbb Z}A_{(m)}z^{-m-1}$, write++$$+A_+(z)=\sum_{m<0}A_{(m)}z^{-m-1},\qquad+A_-(z)=\sum_{m\geq0}A_{(m)}z^{-m-1}.+$$++Here $A_+$ is the creation part and $A_-$ the annihilation part. 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:++$$+:A_1\cdots A_k:=:A_1(:A_2\cdots A_k:):,\qquad :A_1:=A_1.+$$+:::++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. Only the reversed product acquires a parity sign; no homogeneity assumption on $c$ is needed. Hint: combine super-locality and signed skew-symmetry to identify the product and iterate expansions, then apply the formal delta identity.++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.+:::