Vertex Operator Superalgebras and the Majorana Fermion

--- Base version+++ Proposed version@@ -0,0 +1,294 @@+# Amplitudes with local coordinates {@sec:majorana-local-coordinates}++We use the Majorana VOSA of the preceding chapter and the Wick theorem @prop:majorana-wick.++## Local coordinates and the amplitude++Choose distinct finite punctures $z_i$ on the sphere. A local coordinate at $z_i$ is an invertible germ++$$+u=f_i(x),\qquad x=z-z_i,\qquad f_i(0)=0,\quad f_i'(0)\ne0.+$$++Thus $f_i$ maps the global displacement $x$ to the local coordinate $u$. For boundary insertions in the upper half-plane, take $z_i\in\mathbb R$ and boundary-compatible local coordinates. All inserted fields below remain at $z_i$.++For any origin-fixing germ $f$, write++$$+f(x)=\alpha\exp\!\left(\sum_{m\geq1}A_mx^{m+1}\partial_x\right)x,+\qquad+W(f)=\exp\!\left(-\sum_{m\geq1}A_mL_m\right)\alpha^{-L_0}.+$$++The coefficients $A_m$ are determined recursively. Choose a logarithm of $\alpha=f'(0)$ and use compatible lifts under composition. Then++$$+W(f)W(g)=W(g\circ f),\qquad W(f^{-1})=W(f)^{-1}.+$$++On each state of $F$, the positive Virasoro modes lower weight, so the exponential is a finite sum. This is the coordinate-change convention of [Meng–Lan–Gu, Appendix D.1](https://arxiv.org/html/2610.08787).++::: definition [Amplitude with local coordinates] {@def:majorana-local-amplitude}+For states $a_i\in F$, define the normalized chiral vacuum amplitude by++$$+\widehat{\mathcal A}_f(a_1,\ldots,a_n)+=\left\langle+Y(W(f_1)a_1,z_1)\cdots Y(W(f_n)a_n,z_n)+\right\rangle.+$$++The expectation is the vacuum expectation of @prop:majorana-wick, read as a rational function in the ordered insertion points. In particular,+$\widehat{\mathcal A}_f(\mathbf1,\ldots,\mathbf1)=1$.+:::++This is the vacuum-sector specialization of the decorated-sphere amplitude in Appendix D.1, with the outgoing vacuum at infinity. For the primary state $\psi$,++$$+Y(W(f_i)\psi,z_i)=f_i'(0)^{-1/2}\psi(z_i).+$$++For descendants, the positive Virasoro modes in $W(f_i)$ also contribute. The following identities compute these transformed states.++## Coordinate covariance and descendant insertions++Put++$$+g_i=f_i^{-1},\qquad+\varphi_i(u)=z_i+g_i(u),\qquad+\lambda_i=g_i'(0),\qquad s_i(u)^2=g_i'(u).+$$++Here $\varphi_i$ is the insertion map from the local disk to the sphere; it is inverse to the local coordinate function $z\mapsto f_i(z-z_i)$. The square-root branch is fixed by the chosen lift of $f_i$.++::: proposition [Coordinate covariance of the fermion] {@prop:majorana-coordinate-covariance}+For an invertible germ $g(u)=\lambda u+O(u^2)$,++$$+W(g)^{-1}\psi(u)W(g)+=\sqrt{g'(u)}\,\psi(g(u)),+$$++where $\sqrt{g'(0)}=\lambda^{1/2}$ with the chosen logarithm.+:::++::: proof+By @ex:vosa-properties and the primarity of $\psi$,++$$+[L_m,\psi(u)]+=\left(u^{m+1}\partial_u+\frac{m+1}{2}u^m\right)\psi(u).+$$++Write $g(u)=\lambda\exp(v(u)\partial_u)u$, where $v(u)=\sum_{m\geq1}A_mu^{m+1}$, and set $A=\sum_{m\geq1}A_mL_m$. Then++$$+[A,\psi(u)]=\left(v(u)\partial_u+\frac12v'(u)\right)\psi(u).+$$++Let $g_t(u)=\exp(t v(u)\partial_u)u$. The field $F_t(u)=e^{tA}\psi(u)e^{-tA}$ satisfies++$$+\partial_tF_t(u)=\left(v(u)\partial_u+\frac12v'(u)\right)F_t(u),+\qquad F_0(u)=\psi(u).+$$++The chain rule shows that $\sqrt{g_t'(u)}\,\psi(g_t(u))$ satisfies the same equation and initial condition, with square root $1$ at $t=0$. Uniqueness of formal power-series solutions gives++$$+e^A\psi(u)e^{-A}=\sqrt{g_1'(u)}\,\psi(g_1(u)).+$$++Similarly, the $L_0$ commutator gives++$$+\lambda^{L_0}\psi(u)\lambda^{-L_0}+=\lambda^{1/2}\psi(\lambda u).+$$++Since $W(g)^{-1}=\lambda^{L_0}e^A$ and $g=\lambda g_1$, combining these two identities proves the proposition.+:::++::: corollary [Generating transformed descendants] {@cor:majorana-transformed-descendants}+The fields inserted in @def:majorana-local-amplitude satisfy++$$+\Psi_i(u):=Y(W(f_i)e^{uT}\psi,z_i)+=s_i(u)\psi(\varphi_i(u)).+$$++Its coefficients give the transformed descendants at the fixed puncture $z_i$:++$$+[u^m]\Psi_i(u)+=\frac1{m!}Y(W(f_i)T^m\psi,z_i),\qquad m\geq0.+$$+:::++::: proof+Since $W(f_i)=W(g_i)^{-1}$ and $W(g_i)\mathbf1=\mathbf1$, apply @prop:majorana-coordinate-covariance to the vacuum and use creation:++$$+W(f_i)e^{uT}\psi+=s_i(u)e^{g_i(u)T}\psi.+$$++Applying $Y(\,\cdot\,,z_i)$ coefficientwise, translation gives++$$+Y(W(f_i)e^{uT}\psi,z_i)+=s_i(u)Y(\psi,z_i+g_i(u)),+$$++which is the asserted identity.+:::++For example,++$$+\begin{aligned}+[u^0]\Psi_i(u)+&=Y(W(f_i)\psi,z_i)=\lambda_i^{1/2}\psi(z_i),\\+[u^1]\Psi_i(u)+&=Y(W(f_i)T\psi,z_i)\\+&=\lambda_i^{3/2}\partial\psi(z_i)++\frac{g_i''(0)}{2\lambda_i^{1/2}}\psi(z_i).+\end{aligned}+$$++Thus $\Psi_i$ is a way to compute the fields in the amplitude definition. Extracting coefficients at $u=0$ keeps their insertion point at $z_i$.++## Regularized contractions and Pfaffian amplitudes++For states containing several fermions, use a separate local variable for each constituent and normal order in those local variables. The local vacuum contraction is $1/(u-v)$. After the coordinate change, the contraction of the generating fields is+$s_i(u)s_j(v)/(\varphi_i(u)-\varphi_j(v))$.++::: definition [Regularized contraction kernel] {@def:majorana-coordinate-kernel}+Define++$$+K_{ij}(u,v)=+\begin{cases}+\displaystyle\frac{s_i(u)s_j(v)}{\varphi_i(u)-\varphi_j(v)},&i\ne j,\\[3mm]+\displaystyle\frac{s_i(u)s_i(v)}{\varphi_i(u)-\varphi_i(v)}-\frac1{u-v},&i=j.+\end{cases}+$$++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 $\varphi_i(u)=z_i+u$, 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+Write each state as the coefficient of $\prod_\alpha u_\alpha^{r_{i,\alpha}-1/2}$ in $:\psi(u_1)\cdots\psi(u_{k_i}):\mathbf1$, with normal ordering in the local variables. Apply $W(f_i)$ to this state before applying $Y(\,\cdot\,,z_i)$, as required by @def:majorana-local-amplitude. The conjugation identity in @prop:majorana-coordinate-covariance transforms each constituent into $\Psi_i$; the local scalar subtractions are unchanged. Wick's theorem therefore gives pairwise contractions $s_i(u)s_j(v)/(\varphi_i(u)-\varphi_j(v))$. 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++$$+Y(\psi_{-m-\frac12}\mathbf1,z)+=\frac1{m!}\partial_z^m\psi(z).+$$+:::++## Finite-mode calculation++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++   $$+   H_i(u,v)=\frac{\varphi_i(u)-\varphi_i(v)}{u-v}.+   $$++   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++   $$+   K_{ii}(u,v)=\frac{S\varphi_i(0)}{12}(u-v)+   +\text{terms of total degree at least }2.+   $$++   Using $\omega=\tfrac12\psi_{-\frac32}\psi_{-\frac12}\mathbf1$, deduce++   $$+   \widehat{\mathcal A}_f(\mathbf1,\ldots,\omega,\ldots,\mathbf1)+   =\frac{S\varphi_i(0)}{24}.+   $$++   More generally, for three punctures, set $z_{ij}=z_i-z_j$ and verify++   $$+   \widehat{\mathcal A}_f(\omega,\psi,\psi)+   =s_2(0)s_3(0)\left(+   \frac{\lambda_1^2z_{23}}{2z_{12}^2z_{13}^2}+   +\frac{S\varphi_1(0)}{24z_{23}}\right).+   $$++   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.