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

§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 (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 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.

§2 Normal ordering and basic properties

Definition 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 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 5 (Properties of a vertex operator superalgebra)

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

§3 The free Majorana fermion

§Clifford modes and the vacuum space

Definition 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 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: the translates of the vacuum generated by the fermion modes span $F$ $F$ , 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 $\psi_{(-1)}\psi=0$ $0$ , so $:\psi(z)\psi(z):=0$ $0$ .

Proposition 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. See Gaudio, Example 6.1.5 and Bouchard et al., Eq. (4.68).

§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 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 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 $c=\tfrac12$ $\frac{1}{2}$ .

§5 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 $a\in V$ $V$ . In particular, it is stable under $T=L_{-1}$ $L_{- 1}$ . The VOSA is simple if it has no nonzero proper ideal.

Proposition 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.

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

Theorem 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).

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 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 5 to $F_{\frac12}=\mathbb C\psi$ and determine the form $B$ $B$ .

§6 Amplitudes with 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 $\tfrac12$ $\frac{1}{2}$ , 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 $\widehat{\mathcal A}_f(\mathbf1,\ldots,\mathbf1)=1$ $1$ . Normal ordering at puncture $i$ is taken in its local coordinate $u$ $u$ . This is the genus-zero local-coordinate construction of Meng–Lan–Gu, Appendix D.1; the insertion maps here are inverse to the local coordinate functions used there.

Definition 14 (Regularized contraction 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 $K_{ij}(u,v)=-K_{ji}(v,u)$ $- K_{j i} (v, u)$ . For positive half-integers $r,s$ $r, s$ , set

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

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

The subtraction removes the local vacuum contraction. For $f_i(u)=z_i+u$ $u$ , 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 15 (Pfaffian amplitudes)

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 $I$ $I$ . 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 $1$ $1$ . The formula extends multilinearly to arbitrary states in $F$ $F$ .

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 $s_i(u)s_j(v)/(f_i(u)-f_j(v))$ $f_{j} (v))$ . Within a group, local normal ordering subtracts $1/(u-v)$ $v)$ , leaving exactly $K_{ii}$ $K_{ii}$ .

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 $M$ $M$ . 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 $M$ $M$ . 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. The regularization of $K_{ii}$ must be performed in Taylor series before numerical evaluation.

If $a_i$ has weight $h_i$ $h_{i}$ , 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 16 (Coordinate changes and Gaussian amplitudes)

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 $(u-v)^2$ $v)^{2}$ . Deduce the regularity and antisymmetry of $K_{ii}$ $K_{ii}$ , 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.
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 $\omega=\tfrac12\psi_{-\frac32}\psi_{-\frac12}\mathbf1$ $\frac{1}{2} ψ_{- \frac{3}{2}} ψ_{- \frac{1}{2}} 1$ , 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.
Size of the Pfaffian. Show that the least possible weight of a state with $k$ occupied fermion modes is $\tfrac12+\tfrac32+\cdots+(k-\tfrac12)=k^2/2$ $k^{2} /2$ . Deduce the bound on $N$ above.
Generating all amplitudes. For a finite ordered set of mode labels and Grassmann variables $\theta_a$ $θ_{a}$ , 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.

§7 Provenance and license

Author: Chenqi Meng. This independent note is released under Creative Commons Attribution 4.0 International.

The definitions, basic properties, and Majorana example were developed in the VOSA chapter of Introduction to Vertex Operator Algebras, published under CC0. Material previously released under CC0 remains available under those terms. This edition includes the additional treatment of amplitudes with local coordinates.

License · CC BY 4.0 · Peerpedia collaboration terms