Chapter 1 of 3

§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 $\delta(x)=\sum_{m\in\mathbb Z}x^m$ $\sum_{m \in Z} x^{m}$ .

§1.1 Parity and locality

A superspace is a vector space $V=V_{\mathrm{even}}\oplus V_{\mathrm{odd}}$ $V_{odd}$ . 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 $A,B$ $A, B$ , 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 1.1 (Vertex superalgebra)

A vertex superalgebra (VSA) is a superspace $V$ with an even vacuum $\mathbf1$ $1$ , an even translation operator $T$ $T$ , 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 $a,b\in V$ $V$ ,

\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 $a$ $a$ , each mode $a_{(n)}$ has parity $p(a)$ $p (a)$ : 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 $a,b$ $a, b$ , 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 1.2 (Vertex operator superalgebra)

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 $c$ $c$ , called the central charge, and $L_{-1}=T$ $T$ . 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 $V_0=\mathbb C\mathbf1$ $C 1$ .

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 $m>0$ $0$ . A VOSA automorphism is an even linear bijection $g$ fixing $\mathbf1$ and $\omega$ and satisfying $gY(a,z)b=Y(ga,z)gb$ $Y (g a, z) g b$ .

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.

An ideal of a VOSA $V$ is a parity-graded subspace $I$ satisfying $Y(a,z)I\subseteq I((z))$ for every $a\in V$ $V$ . In particular, it is stable under $T=L_{-1}$ $L_{- 1}$ . The VOSA is simple if it has no nonzero proper ideal.

§1.2 Normal ordering and basic properties

Definition 1.3 (Super normal ordering)

For a field $A(z)=\sum_{m\in\mathbb Z}A_{(m)}z^{-m-1}$ $\sum_{m \in Z} A_{(m)} z^{- m - 1}$ , 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 $A,B$ $A, B$ , 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 $p(b)$ $p (b)$ , the states $a,b$ are parity-homogeneous; extend the identities by linearity.

Exercise 1.4 (Properties of a vertex superalgebra)

Let $(V,Y,\mathbf1,T)$ be a VSA. Prove the following identities.

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.
Signed skew-symmetry.

$Y(a,z)b=(-1)^{p(a)p(b)}e^{zT}Y(b,-z)a.$
Super Jacobi identity. For parity-homogeneous $a,b$ and any $c\in V$ $V$ ,

$\begin{aligned} &x^{-1}\delta\!\left(\frac{z-w}{x}\right)Y(a,z)Y(b,w)c\\ &\quad-(-1)^{p(a)p(b)}x^{-1}\delta\!\left(\frac{w-z}{-x}\right)Y(b,w)Y(a,z)c\\ &\qquad=z^{-1}\delta\!\left(\frac{w+x}{z}\right)Y(Y(a,x)b,w)c. \end{aligned}$

The binomials in the three delta kernels are expanded in nonnegative powers of $w$ $w$ , $z$ $z$ , and $x$ $x$ , 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.
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.
Zero-mode superderivations.

$[a_{(0)},Y(b,z)]_{\mathrm s}=Y(a_{(0)}b,z), \qquad a_{(0)}\mathbf1=0.$

Equivalently, for $d\in V$ $V$ ,

$a_{(0)}\bigl(Y(b,z)d\bigr) =Y(a_{(0)}b,z)d +(-1)^{p(a)p(b)}Y(b,z)a_{(0)}d.$
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 1.5 (Properties of a vertex operator superalgebra)

Let $(V,Y,\mathbf1,\omega)$ be a VOSA with the parity-compatible grading of Definition 1.2. Prove the following properties.

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 $b\in V_k$ $V_{k}$ ,

$a_{(n)}b\in V_{h+k-n-1},\qquad TV_h\subseteq V_{h+1}.$
Virasoro covariance. For every $a\in V$ and $m\in\mathbb Z$ $Z$ ,

$[L_m,Y(a,z)] =\sum_{j\geq0}\binom{m+1}{j}z^{m+1-j}Y(L_{j-1}a,z).$

No parity sign appears because $\omega$ is even. In particular, for a primary state $a$ of weight $h$ $h$ ,

$[L_m,Y(a,z)] =\bigl(z^{m+1}\partial_z+(m+1)hz^m\bigr)Y(a,z).$
Parity and the even VOA. The linear map $\sigma$ $σ$ , equal to $+1$ on $V_{\mathrm{even}}$ and $-1$ on $V_{\mathrm{odd}}$ $V_{odd}$ , is a VOSA automorphism. Its fixed-point space $V_{\mathrm{even}}$ $V_{even}$ , with the restricted fields and the same vacuum and conformal vector, is an ordinary VOA of the same central charge. Moreover,

$\sigma=e^{2\pi iL_0}.$
Weight-one-half fermions. Assume in addition that $V$ is of CFT type. For $a,b\in V_{\frac12}$ $V_{\frac{1}{2}}$ , 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.

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