# Vertex Operator Superalgebras and the Majorana Fermion

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).

## 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.
:::

## 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: the translates of the vacuum generated by the fermion modes span $F$, and the Clifford relations supply 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. 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

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.

::: 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

$$
F_0=\mathbb C\mathbf1,\quad F_{\frac12}=\mathbb C\psi,
\quad F_1=0,\quad F_{\frac32}=\mathbb C T\psi,
\quad F_2=\mathbb C\omega.
$$

Write $\operatorname{Vir}(\omega)$ for the vertex subalgebra generated by $\omega$, and $L(c,h)$ for the irreducible highest-weight Virasoro module of central charge $c$ and highest weight $h$. The vacuum module $L(c,0)$ has its simple Virasoro VOA structure. The even part of $F$ has the following identification:

::: theorem [The even fermion algebra] {@thm:majorana-ising}
The even subalgebra of the Majorana VOSA is the Ising VOA:

$$
F_{\mathrm{even}}=\operatorname{Vir}(\omega)
\cong L\!\left(\frac12,0\right).
$$
:::

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 $B$.
:::

## Amplitudes with local coordinates {@sec:majorana-local-coordinates}

Consider $n$ ordered punctures on the Riemann sphere. Choose a global coordinate in which their positions $z_i$ are finite and distinct, and holomorphic insertion maps

$$
f_i(u)=z_i+\lambda_i u+O(u^2),\qquad \lambda_i\ne0.
$$

Here $u$ is the local coordinate at the $i$th puncture: $f_i$ maps the local disk into the sphere. Fix a square-root germ $s_i(u)^2=f_i'(u)$ at each puncture. Since $\psi$ is primary of weight $\tfrac12$, its field in that coordinate is

$$
\Psi_i(u)=s_i(u)\psi(f_i(u)).
$$

Write $\widehat{\mathcal A}_f(a_1,\ldots,a_n)$ for the chiral amplitude of states $a_i\in F$ inserted in these local coordinates, normalized by $\widehat{\mathcal A}_f(\mathbf1,\ldots,\mathbf1)=1$. Normal ordering at puncture $i$ is taken in its local coordinate $u$. This is the genus-zero local-coordinate construction of [Meng–Lan–Gu, Appendix D.1](https://arxiv.org/html/2610.08787); the insertion maps here are inverse to the local coordinate functions used there.

::: definition [Regularized contraction kernel] {@def:majorana-coordinate-kernel}
Define

$$
K_{ij}(u,v)=
\begin{cases}
\displaystyle\frac{s_i(u)s_j(v)}{f_i(u)-f_j(v)},&i\ne j,\\[3mm]
\displaystyle\frac{s_i(u)s_i(v)}{f_i(u)-f_i(v)}-\frac1{u-v},&i=j.
\end{cases}
$$

These are Taylor series at $(0,0)$ and satisfy $K_{ij}(u,v)=-K_{ji}(v,u)$. For positive half-integers $r,s$, set

$$
M_{(i,r),(j,s)}
=[u^{r-\frac12}v^{s-\frac12}]K_{ij}(u,v).
$$

Thus $M^{\mathsf T}=-M$. The brackets extract Taylor coefficients, including the factorial normalization of derivatives.
:::

The subtraction removes the local vacuum contraction. For $f_i(u)=z_i+u$, the entire diagonal block vanishes, recovering the usual rule that contractions occur only between different normal-ordered factors. For a nonlinear coordinate, its regular remainder may be nonzero. These diagonal blocks encode the finite change of normal ordering.

::: proposition [Pfaffian amplitudes] {@prop:majorana-coordinate-pfaffian}
Let

$$
a_i=\psi_{-r_{i,1}}\cdots\psi_{-r_{i,k_i}}\mathbf1,
\qquad r_{i,1}>\cdots>r_{i,k_i}>0.
$$

Concatenate the occupied labels $(i,r_{i,\alpha})$ in puncture order, preserving the displayed order within each state, to obtain an ordered list $I$. Then

$$
\widehat{\mathcal A}_f(a_1,\ldots,a_n)
=\operatorname{Pf}M[I,I].
$$

For an odd number of occupied modes the amplitude is zero; for the empty list it is $1$. The formula extends multilinearly to arbitrary states in $F$.
:::

::: proof
Represent each state by a locally normal-ordered group of fermions, assigning a separate variable to each constituent. Wick's theorem gives pairwise contractions $s_i(u)s_j(v)/(f_i(u)-f_j(v))$. Within a group, local normal ordering subtracts $1/(u-v)$, leaving exactly $K_{ii}$.

In each pairing, every variable occurs in exactly one kernel. Extracting the coefficient prescribed by each occupied mode therefore gives a product of entries of $M$. The signed sum over pairings is its Pfaffian. In particular, the coefficient convention agrees with

$$
Y(\psi_{-m-\frac12}\mathbf1,z)
=\frac1{m!}\partial_z^m\psi(z).
$$
:::

For a fixed finite set of modes, compute the required Taylor coefficients of $K_{ij}$ once and store $M$. Each occupation-basis amplitude then requires only a selected submatrix. A Pfaffian of order $N$ can be evaluated by pivoted antisymmetric elimination in $O(N^3)$ arithmetic operations and $O(N^2)$ storage, without enumerating pairings; see [Wimmer](https://arxiv.org/abs/1102.3440). The regularization of $K_{ii}$ must be performed in Taylor series before numerical evaluation.

If $a_i$ has weight $h_i$, fermionic exclusion gives

$$
h_i\geq\frac{k_i^2}{2},\qquad
N=\sum_i k_i\leq\sum_i\lfloor\sqrt{2h_i}\rfloor.
$$

For general states, apply the formula to their occupation-basis expansions. The cost of producing every entry of a tensor with $D$ states at each puncture still includes its $D^n$ output entries. All amplitudes here belong to the Majorana vacuum block; an overall geometric vacuum factor and amplitudes involving spin fields require separate data.

::: exercise [Coordinate changes and Gaussian amplitudes] {@ex:majorana-local-coordinates}
1. **Regularity.** Put

   $$
   H_i(u,v)=\frac{f_i(u)-f_i(v)}{u-v}.
   $$

   Show that $H_i$ is an invertible Taylor series and that $s_i(u)s_i(v)-H_i(u,v)$ is divisible by $(u-v)^2$. Deduce the regularity and antisymmetry of $K_{ii}$, and use

   $$
   K_{ii}(u,v)=\frac{s_i(u)s_i(v)-H_i(u,v)}{(u-v)H_i(u,v)}
   $$

   to compute its Taylor coefficients without subtracting poles numerically.

2. **The Schwarzian term.** Define

   $$
   Sf=\frac{f'''}{f'}-\frac32\left(\frac{f''}{f'}\right)^2.
   $$

   Prove

   $$
   K_{ii}(u,v)=\frac{Sf_i(0)}{12}(u-v)
   +\text{terms of total degree at least }2.
   $$

   Using $\omega=\tfrac12\psi_{-\frac32}\psi_{-\frac12}\mathbf1$, deduce

   $$
   \widehat{\mathcal A}_f(\mathbf1,\ldots,\omega,\ldots,\mathbf1)
   =\frac{Sf_i(0)}{24}.
   $$

   More generally, for three punctures, set $z_{ij}=z_i-z_j$ and verify

   $$
   \widehat{\mathcal A}_f(\omega,\psi,\psi)
   =s_2(0)s_3(0)\left(
   \frac{\lambda_1^2z_{23}}{2z_{12}^2z_{13}^2}
   +\frac{Sf_1(0)}{24z_{23}}\right).
   $$

   Identify the pairings responsible for the two terms.

3. **Size of the Pfaffian.** Show that the least possible weight of a state with $k$ occupied fermion modes is $\tfrac12+\tfrac32+\cdots+(k-\tfrac12)=k^2/2$. Deduce the bound on $N$ above.

4. **Generating all amplitudes.** For a finite ordered set of mode labels and Grassmann variables $\theta_a$, prove

   $$
   \exp\!\left(\frac12\sum_{a,b}\theta_aM_{ab}\theta_b\right)
   =\sum_{I\text{ even}}\operatorname{Pf}M[I,I]\,\theta_I,
   $$

   where $I$ runs through subsets in the inherited order and $\theta_I$ is the corresponding ordered product. Explain how $M$ determines all occupation-basis amplitudes without storing the full tensor.
:::

## Provenance and license

Author: Chenqi Meng. This independent note is released under [Creative Commons Attribution 4.0 International](https://creativecommons.org/licenses/by/4.0/).

The definitions, basic properties, and Majorana example were developed in the VOSA chapter of [Introduction to Vertex Operator Algebras](https://peerpedia.dev/submissions/261030.6b9c1cee3c0f/read/?chapter=vertex-operator-superalgebras), published under CC0. Material previously released under CC0 remains available under those terms. This edition includes the additional treatment of amplitudes with local coordinates.