Chapter 8 of 11

§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.

§8.1 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 8.1 (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 the truncation, vacuum, creation, and translation axioms of Definition 2.1. 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. Locality is replaced by super-locality: 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 8.2 (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, $L_{-1}=T$ $T$ , 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 $V_0=\mathbb C\mathbf1$ $C 1$ .

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.

§8.2 Normal ordering and basic properties

Definition 8.3 (Super normal ordering)

Retain the split $A=A_++A_-$ from Definition 2.12, with $A_+$ containing the modes $A_{(n)}$ for $n<0$ $0$ . 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.

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 8.4 (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, as in Proposition 2.10. Only the reversed product acquires a parity sign; no homogeneity assumption on $c$ is needed. Adapt the proof of Proposition 2.10 using super-locality.
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 8.5 (Properties of a vertex operator superalgebra)

Let $(V,Y,\mathbf1,\omega)$ be a VOSA with the parity-compatible grading of Definition 8.2. 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.

§8.3 The free Majorana fermion

§Clifford modes and the vacuum space

Definition 8.6 (Fermionic vacuum space)

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 $n\geq0$ $0$ , 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 $(-1)^{i-1}$ $(- 1)^{i - 1}$ . 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 $z$ $z$ : in vertex-algebra notation, $\psi_{(n)}=\psi_{n+\frac12}$ $ψ_{n + \frac{1}{2}}$ . Its creation part consists of $r<0$ and its annihilation part of $r>0$ $0$ .

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 $s\bmod2$ $2$ . The strict inequalities express the fact that a fermionic creation mode cannot occur twice.

§Fields and conformal structure

Proposition 8.7 (Majorana vertex superalgebra)

There is a unique vertex superalgebra structure on $F$ with $Y(\psi,z)=\psi(z)$ $ψ (z)$ . 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 $\psi(z)\mathbf1\in\psi+zF[[z]]$ $z F [[z]]$ . The superalgebra reconstruction theorem applies, just as in Proposition 7.2, 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 $\psi_{(-1)}\psi=0$ $0$ , so $:\psi(z)\psi(z):=0$ $0$ .

Proposition 8.8 (Majorana conformal vector)

The even vector

\omega=\frac12\psi_{-\frac32}\psi_{-\frac12}\mathbf1

makes $F$ a VOSA of CFT type with central charge $c=\tfrac12$ $\frac{1}{2}$ . 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 $\tfrac12$ $\frac{1}{2}$ , 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 $\psi_s$ $ψ_{s}$ , use

[\psi_a\psi_b,\psi_s] =\psi_a\{\psi_b,\psi_s\}-\{\psi_a,\psi_s\}\psi_b.

The two Kronecker deltas yield $[L_m,\psi_s]=-(s+m/2)\psi_{m+s}$ $m /2) ψ_{m + s}$ . Hence $L_{-1}=T$ $T$ , 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 $L_{-1}\omega=T\omega$ $T ω$ , give

[L_m,L_n]=(m-n)L_{m+n} +\frac{m^3-m}{24}\delta_{m+n,0}\operatorname{id}.

Thus $c=\tfrac12$ $\frac{1}{2}$ . Finally $L_0\psi=\tfrac12\psi$ and $L_n\psi=0$ for $n>0$ $0$ , proving primarity.

This is the free Majorana fermion VOSA, also called the neutral free fermion. The construction is over $\mathbb C$ $C$ , as in the rest of the note. See Gaudio, Example 6.1.5 and Bouchard et al., Eq. (4.68).

§8.4 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 $|z_1|>\cdots>|z_n|$ $∣ z_{n} ∣$ . In particular,

\langle\psi(z)\psi(w)\rangle=\frac1{z-w}.

Proposition 8.9 (Fermionic Wick theorem)

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 $1$ $1$ .

Proof.

With the creation/annihilation convention already fixed, $\varepsilon\psi_+(z)=0$ and $\psi_-(z)\mathbf1=0$ $0$ . 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 $G_n(z_1,\ldots,z_n)=\langle\psi(z_1)\cdots\psi(z_n)\rangle$ $⟨ ψ (z_{1}) \dots ψ (z_{n})⟩$ . 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 $G_1=0$ $0$ , 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.

Every field of $F$ is a linear combination of normal-ordered derivatives of $\psi$ $ψ$ . 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 $F$ $F$ , including the conformal vector and its descendants.

Exercise 8.10 (Stress-tensor two-point function)

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 $c=\tfrac12$ $\frac{1}{2}$ .

§8.5 Simplicity and the Ising VOA

Proposition 8.11 (Simplicity of the free fermion)

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 $\mathbf1$ $1$ . Creation modes then generate all of $F$ $F$ .

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 8.12 (The even fermion algebra)

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. The full $F$ also contains the odd field $\psi$ of weight $\tfrac12$ $\frac{1}{2}$ , so it is a VOSA extension of the Ising VOA, not the same algebra.

Exercise 8.13 (Parity and fermionic products)

Verify directly that $\sigma(\psi)=-\psi$ preserves the Clifford relations and fixes $\omega$ $ω$ . Identify $F_{\mathrm{even}}$ in the oscillator basis.
Compute $\psi_{(-2)}\psi$ and verify $:(\partial_z\psi(z))\psi(z):=2Y(\omega,z)$ $2 Y (ω, z)$ .
Apply Exercise 8.5 to $F_{\frac12}=\mathbb C\psi$ and determine the form $B$ $B$ .

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