Vertex Operator Superalgebras and the Majorana Fermion

--- Base version+++ Proposed version@@ -286,145 +286,4 @@ 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.+Amplitudes with specified local coordinates are treated in @sec:majorana-local-coordinates.