Vertex Operator Superalgebras and the Majorana Fermion

--- Base version+++ Proposed version@@ -0,0 +1,430 @@+# The 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++::: 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.