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Use the curvature sign convention implicit in part (c): the Jacobi field equation is $D_t^{2}J+K_{\dot{\gamma }}J=0$, so the operator in a positively curved normal direction has positive eigenvalue. With the alternative convention $R_{\mathrm{std}}(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z$ used for the catalog's Jacobi curvature operator, the paper's tensor is $R=-R_{\mathrm{std}}$, and its operator is $K_v(x)=R_{\mathrm{std}}(x,v)v$. This fixes the sign throughout Q1.

Let $r(A,B,C,D)=⟨R(A,B)C,D⟩$. Pair interchange for the Riemann curvature tensor givesThus $K_v$ is self-adjoint with respect to the Riemannian metric. The curvature sign does not affect this conclusion. The spectral theorem for real symmetric matrices therefore supplies the real orthonormal eigenbasis used in part (b). Also $K_{v}v=0$, so the tangent direction is a zero-eigenvalue direction.

Write $D_t={\nabla }_{\dot{\gamma }}$ and let $P_t:T_{p}M\to T_{\gamma (t)}M$ denote parallel transport. Since $\gamma$ is a geodesic, $D_t\dot{\gamma }=0$, while the transported vectors satisfy $D_t{e}_i(t)=0$. The locally symmetric Riemannian manifold condition is $\nabla R=0$. Hence the Levi-Civita connection product rule yieldsBoth this vector field and ${\lambda }_i{e}_i(t)$ are parallel and agree at zero. Uniqueness of parallel transport therefore provesThe eigenvalues are independent of $t$. The original PDF has $K_{\dot{\gamma }(t)}$ here; the converted TeX omits the dot, which would incorrectly put a point rather than a tangent vector in the operator's subscript.

Expand a Jacobi field in the parallel orthonormal eigenbasis: $J(t)={\sum }_i{j}_i(t)e_i(t)$. Part (b) reduces its equation to the scalar equations $j_i^{\prime \prime }+{\lambda }_i{j}_i=0$. DefineFor arbitrary initial values $J(0)={\sum }_i{a}_i{e}_i$ and $D_{t}J(0)={\sum }_i{b}_i{e}_i$, the complete solution isA point at parameter $T>0$ is a conjugate point of $p$ precisely when a nonzero Jacobi field vanishes at both $0$ and $T$. The first condition makes every $a_i=0$. The second requires $b_i{S}_{{\lambda }_i}(T)=0$ for every $i$, with at least one nonzero $b_i$. For $T>0$, the factors for zero and negative eigenvalues never vanish; positive eigenvalues give exactly the sine zeros. ThereforeThe corresponding points are $\gamma (T)$. If several positive eigenspaces have a zero at the same parameter, the conjugate multiplicity is the sum of their dimensions. This also shows that a locally symmetric Riemannian manifold has no conjugate points along a given geodesic when that geodesic's Jacobi curvature operator has no positive eigenvalues.

Parametrize the nonconstant closed geodesic at unit speed on $[0,L]$. Its parallel transport around the loop fixes the tangent $\dot{\gamma }(0)$. Since the manifold is orientable, it preserves orientation on the full tangent space, and hence has determinant one on the normal space. This normal space has odd dimension because the manifold has even dimension.

An odd-dimensional special orthogonal transformation has a fixed vector. Indeed, nonreal eigenvalues of a real orthogonal transformation occur in conjugate pairs with product one; the real eigenvalues are $\pm 1$, and determinant one makes the number of $-1$ eigenvalues even. Odd dimension therefore requires a $+1$ eigenvalue. Transport a unit fixed normal vector around the geodesic to obtain a periodic parallel field $V$, with $D_{t}V=0$ and $V\perp \dot{\gamma }$.

The variation ${\gamma }_s(t)={\exp }_{\gamma (t)}(sV(t))$ is defined uniformly for small $s$, because the image of the original loop is compact. Periodicity gives closed curves and a homotopy through loops. With the standard positive-sectional-curvature Riemannian index form, the second variation of geodesic energy isThe first variation vanishes for a closed geodesic, since the endpoint terms cancel. Therefore $E({\gamma }_s)<E({\gamma }_0)=L/2$ for sufficiently small nonzero $s$. The Cauchy-Schwarz inequality givesThus the closed geodesic is homotopic to a strictly shorter closed curve. This proves the instability of a closed geodesic in positive even-dimensional curvature; completeness of the manifold is unnecessary for this compact-loop variation. Constant loops are excluded by the usual nonconstant meaning of a closed geodesic in this assertion.

Interpret $\theta$ as a continuous finite-dimensional complex group representation, so its values lie in ${\mathrm{GL}}_{\mathbb{C}}(V)$, and interpret approximation in the uniform norm. Assume the usual Hausdorff convention for the compact topological group. The following argument proves the required form of the Peter-Weyl theorem without first assuming that $G$ is a Lie group.

Start with normalized Haar measure. It is both left- and right-invariant on a compact group. Average any Hermitian inner product to make a finite-dimensional representation unitary. The left regular action $L_{a}h(x)=h(a^{-1}x)$ is already unitary on $L^2(G)$.

Choose a continuous nonnegative function $k$ supported in a small symmetric identity neighbourhood, with $k(z^{-1})=k(z)$ and $\int k=1$. Such functions are obtained from a nonzero continuous bump near the identity, symmetrizing and normalizing. DefineThe convolution operator $T_k$ commutes with left translations. Its continuous kernel is Hermitian, hence it is self-adjoint. It is compact on $L^2(G)$: approximate the continuous kernel uniformly by finite sums of separated functions of $x$ and $y$, using the Stone-Weierstrass theorem on the compact product. The corresponding operators have finite rank and converge in operator norm. It also maps into $C(G)$ and satisfiesby the Cauchy-Schwarz inequality; continuity follows from the uniform continuity of the translated kernel.

The spectral theorem for compact self-adjoint operators decomposes the orthogonal complement of its kernel into finite-dimensional eigenspaces with nonzero eigenvalues. Every such eigenspace is left-translation invariant, and all its elements are continuous, since $h=T_{k}h/\lambda$ there. If $W\subset C(G)$ is one of these invariant spaces, write $\rho (a)=L_a{|}_W$. Evaluation at the identity givesThus $h$ is a matrix coefficient of the dual representation. This identifies each finite spectral sum with finite-dimensional representation data.

To get uniform rather than just $L^2$ approximation, let $S_N$ project onto increasing finite sums of the nonzero eigenspaces. For $f\in C(G)$, $T_{k}f$ is orthogonal to the kernel, so $S_N{T}_{k}f\to T_{k}f$ in $L^2$. Applying the displayed smoothing bound givesMeanwhile, choosing the support of $k$ sufficiently small makes $T_{k}f$ uniformly close to $f$, by uniform continuity under small translations. Since $\Vert T_k{\Vert }_{C\to C}\le 1$, it also makes $T_k^{2}f$ uniformly close to $f$. Every $T_k{S}_N{T}_{k}f$ belongs to a finite sum of invariant eigenspaces, hence is a finite sum of matrix coefficients. This completes the convolution proof of uniform Peter-Weyl approximation.

A coefficient $\ell (\theta (g)v)$ equals $\mathrm{tr}(\alpha \theta (g))$ for the rank-one endomorphism $\alpha =v\otimes \ell$. Finite sums of coefficients can be combined using a direct-sum representation and a block-diagonal $\alpha$. Therefore the conclusion has exactly the requested form:

For a class function $f$, apply conjugation averaging on a compact group,It fixes $f$ and is a contraction in the uniform norm, so averaging an approximation preserves its error bound. A trace coefficient averages to $\mathrm{tr}(\overline{\alpha }\theta (g))$, whereThis endomorphism commutes with the representation. A finite-dimensional unitary representation splits into irreducibles by repeatedly taking invariant orthogonal complements. On each irreducible diagonal block, Schur lemma makes its average a scalar multiple of the identity, namely its trace divided by the block dimension. Off-diagonal blocks do not contribute to the total trace. Thus the averaged coefficient is a finite linear combination of irreducible characters. Consequently irreducible complex characters span a uniformly dense subspace of the continuous class functions.

The within-group and between-group scatter decomposition separates noise from group separation: $W$ measures variation about each group sample mean, and $B$ measures separation of those means about the overall sample mean. In linear discriminant analysis, seek a projection $a^{T}x$ maximizing between-group relative to within-group variation,Normalize by $a^{T}Wa=1$; a Lagrange multiplier calculation gives the generalized eigenvalue problem $Ba=\lambda Wa$. The leading canonical discriminant directions have the largest eigenvalues, and subsequent directions are orthogonal in the $W$ inner product. There are at most $\min (p,g-1)$ nonzero separating directions. Projecting observations onto these directions gives a low-dimensional display of group separation.

For classification, estimate the common covariance matrix by $S_p=W/(n-g)$ and use the linear discriminant analysis scoreswhere ${\pi }_{\nu }$ are class prior probabilities. Assign to the largest score. The same within-group scatter sets the noise scale, while between-group scatter selects the directions in which the group means separate most strongly.

princomp(): principal component analysis. Supply observations in rows and numeric variables in columns. After centering, let $Z$ denote the resulting matrix and $C=Z^{T}Z/n$ its sample covariance matrix, with an explicitly stated normalization. WriteThe first direction maximizes the Rayleigh quotient $q^{T}Cq$ over unit vectors; successive directions maximize it subject to orthogonality to the earlier directions. The principal component scores are $z_{ij}=q_j^T(x_i-\bar{x})$, with variances ${\lambda }_j$ and zero pairwise sample covariance. Zero covariance alone does not imply independence. Each principal component loading describes a direction in variable space; its sign is arbitrary, and repeated eigenvalues leave the basis inside the corresponding eigenspace arbitrary.

The function returns principal component loadings, component standard deviations sdev and, when requested, principal component scores. By default it uses a covariance matrix; cor=T instead performs principal component analysis on a correlation matrix, standardizing nonconstant variables to variance one. This choice matters when variables have different measurement units. These interface details are documented in the S-Plus help for princomp.

A scree plot shows the ordered eigenvalues; a plot of the first two principal component scores can reveal groups, unusual observations or a dominant gradient. The first $k$ components explain ${\sum }_{j\le k}{\lambda }_j/{\sum }_j{\lambda }_j$ of the total variance. Reconstructing each centered observation by its orthogonal projection onto these directions minimizes average squared reconstruction error among $k$-dimensional linear subspaces. Thus PCA gives an optimal linear compression for squared reconstruction error, with scaling and the retained dimension chosen explicitly. It is an unsupervised learning method, rather than a classification rule fitted to supplied labels.

tree(): classification trees and regression trees. A call such as tree(response ~ predictors, data=dat) fits a recursive partitioning model. At a current node, test numerical predictors against candidate thresholds, or divide the levels of a categorical predictor into two sets. Select a split that most improves the fitting criterion, then repeat within its children. Numerical splits give axis-aligned regions. A categorical response produces a classification tree; a numeric response produces a regression tree, as specified in the S-Plus tree documentation.

For a regression tree, the constant prediction in a terminal region is its sample mean, minimizing the sum of squared residuals there. For a classification tree, estimate class probabilities by the observed proportions in a leaf and predict a most probable class under equal misclassification costs. The classification-tree deviance of a node iswith $0\log 0=0$; splitting maximizes the decrease from parent to children. This fitting criterion differs from the number of wrong classifications. The middle sketches show the same three-region partition as a binary tree: first split on variable one, then split the right-hand branch on variable two.

Growing to small leaves can overfit. Use cost-complexity tree pruning, balancing fitted loss $R(T)$ against leaf count,and select complexity by cross-validation. Plotting the fitted tree and annotating nodes displays the rules; prediction routes each new observation through the fitted splits. A tree gives an interpretable, piecewise-constant prediction rule; training fit alone does not measure prediction accuracy.

The rpart() alternative. This also fits classification and regression trees by recursive partitioning, with squared-error splitting for ordinary numeric responses and, by default for classification, Gini impurity $1-{\sum }_r{p}_r^2$. Classification settings can instead use information-based splitting and specify prior probabilities and a loss matrix. It supports surrogate splits to approximate a node's primary left/right assignment when the primary predictor is missing. Its complexity-parameter path and cross-validated errors support cost-complexity tree pruning; for example, select a complexity parameter and apply prune(fit, cp=...). The maintainer's rpart documentation describes the criteria and the distinction in handling surrogate predictors. The partition and tree sketches apply to either implementation; the fitted tree need not be the same because splitting criteria and controls can differ.

cmdscale(): classical multidimensional scaling. Its input is a dissimilarity matrix or stored distance object. It outputs coordinates for the observations in a chosen dimension, usually two for visualization. Let $D^{(2)}$ contain the squared entries of $D$, and putFor centered Euclidean coordinates $X$, the distance identity $d_{ij}^2=\Vert x_i{\Vert }^2+\Vert x_j{\Vert }^2-2x_i^T{x}_j$ gives $G=X{X}^T$: multiplying by $H$ removes the row and column norm terms. Diagonalize this Gram matrix and retain its $k$ largest positive eigenvalues. The coordinates areThus classical scaling exactly recovers Euclidean distances when all positive eigenvalues are retained. Reducing dimension approximates the centered Gram matrix; it does not in general minimize squared discrepancies in the original distances. Negative eigenvalues identify a failure of the Euclidean distance matrix criterion, so their size matters when interpreting a display. The historical S-Plus cmdscale help describes the dimension argument, eigenvalue output and optional additive adjustment to dissimilarities.

Plot the two coordinate columns on equally scaled axes and label observations. Nearby plotted points suggest small dissimilarities only insofar as the dimension reduction fits the input well. The bottom sketches give an exact Euclidean example: a configuration and its independently recovered two-dimensional coordinates have identical pairwise distances, despite different coordinate axes. Classical scaling converts distances into geometry; translations, rotations and reflections do not change the represented distances. For Euclidean distances between observations, its coordinates agree with the corresponding principal component scores up to these harmless choices of axes.

Use a mostly-plus target metric tensor and the standard open-string mode expansion convention ${\alpha }_0^{\mu }=\sqrt{2{\alpha }^{\prime }}{p}^{\mu }$. The integer string oscillator level is $N={\sum }_{n\ge 1}{\alpha }_{-n}\cdot {\alpha }_n$, whose eigenvalues are nonnegative integers. Then $L_0=N+{\alpha }^{\prime }{p}^2$, so the string mass-shell condition givesThe ground state is a tachyon, and the level-one state is massless. The physical string state conditions in old covariant string quantization areOnly the positive-mode Virasoro constraints annihilate a ket; their negative-mode partners annihilate the corresponding bra. Imposing both signs on every ket would contradict the nonzero Virasoro central extension.

The central charge comes from normal ordering the quadratic oscillator expressions. Classically their Poisson brackets have no central term. Quantum mechanically, moving annihilation operators through creation operators produces double contractions. From the oscillator commutator one obtains $[L_m,{\alpha }_n^{\mu }]=-n{\alpha }_{m+n}^{\mu }$, which gives the noncentral part $(m-n)L_{m+n}$.

The coefficient is fixed by the free-boson Virasoro central term. On a formal zero-momentum vacuum, for $m>0$,The two contraction pairings giveHere ${\eta }_{\mu \nu }{\eta }^{\mu \nu }=D$; the time coordinate contributes one, just as each spatial coordinate does. Comparing with the Virasoro algebra provesThis counts covariant target coordinates. It is distinct from the $D-2$ physical transverse polarizations. Including the central charge of reparameterization ghosts gives $c_{\mathrm{total}}=D-26$; cancellation of the worldsheet Weyl anomaly gives the usual critical dimension of the bosonic string $D=26$.

A general level-one state isThe zero-mode constraint requires $k^2=0$. Since $[L_1,{\alpha }_{-1}^{\mu }]={\alpha }_0^{\mu }$, the remaining nontrivial positive-mode constraint is $k\cdot ϵ=0$; all $L_m$ with $m\ge 2$ act trivially at this level. Thus the polarization vector is transverse to a nonzero null momentum.

There is also a null string state $L_{-1}|0;k⟩=\sqrt{2{\alpha }^{\prime }}k\cdot {\alpha }_{-1}|0;k⟩$. It is physical when $k^2=0$ and orthogonal to every physical polarization. The null-state quotient of a string therefore identifiesChoose $k=(E,E,0,\dots ,0)$. Transversality sets ${ϵ}^1={ϵ}^0$, and the null-state identification removes this common component. The remaining $D-2$ spatial components have positive norm. Hence the physical level-one states form a massless vector with $D-2$ polarizations, or 24 at the critical dimension. The removed component is gauge redundancy. States violating the constraints, including independent timelike negative-norm polarizations, are unphysical; physical null states are removed by the quotient rather than counted as additional particles.

Start with the Clifford algebra $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }$. For the dimension count, complexify and rescale the timelike matrix so that the generators obey the Euclidean complex relation $\{{\gamma }_a,{\gamma }_b\}=2{\delta }_{ab}$. This changes neither the complex representation dimension nor its decomposition under the complexified even algebra. In $D=2k$, pair the generators intoDirect substitution gives $\{a_i,a_j\}=\{b_i,b_j\}=0$ and $\{a_i,b_j\}={\delta }_{ij}$. The commuting operators $n_j=b_j{a}_j$ satisfy $n_j^2=n_j$, so their occupation eigenvalues are zero or one. An irreducible module contains an all-zero occupation state $|0⟩$, obtained by lowering the occupied entries of a simultaneous eigenvector. The statesare nonzero and independent by their distinct joint occupation eigenvalues. They span an invariant module; irreducibility makes this the full module. Conversely this Fock construction supplies a representation on exactly these $2^k$ states. Products of occupation projectors and raising/lowering operators give its matrix units, so the generated algebra is the full matrix algebra. Therefore the minimal complex Dirac spinor has

This module is irreducible under the full Clifford algebra, but the Spinor representation of the Lorentz group is generated by the even bilinears ${\Sigma }^{\mu \nu }=\frac{1}{4}[{\gamma }^{\mu },{\gamma }^{\nu }]$. The chirality matrix in the Euclideanized basis isIts two eigenspaces therefore remain invariant under Lorentz transformations. Each has half the Fock states, since any gamma matrix bijects them. The even algebra connects every pair of occupation states of the same parity and has no further invariant subspaces within either one. HenceThese are the irreducible Weyl spinors. Globally they are ordinary representations of $\mathrm{Spin}(D-1,1)$, the double cover; calling them representations of $SO(D-1,1)$ uses the conventional projective spinor terminology.

For $D=2k+1$, adjoin ${\gamma }_{2k+1}=\pm {\gamma }_{\ast }$. It anticommutes with the old matrices and gives a Clifford module of the same dimension $2^k$. The new even generators $[{\gamma }_a,{\gamma }_{2k+1}]$ interchange the two old chirality eigenspaces, so the odd-dimensional Lorentz spinor is irreducible of complex dimension $2^k$, with no Weyl splitting. The product of all odd-dimensional gamma matrices is central and acts as a scalar; it cannot be a new anticommuting chirality operator. Its two signs distinguish the two complex odd-Clifford modules, whose restrictions to Spin group are equivalent. Majorana spinor reality conditions, when the signature permits them, are additional conditions and should not be confused with complex chirality.

For the reduction, take the massless zero modes on a flat seven-torus with no flux or projections. The eleven-dimensional fields are the metric $G_{MN}$, a three-form $A_{MNP}$ and a Majorana gravitino ${\Psi }_M$. Split $M=(\mu ,i)$ with $\mu =0,\dots ,3$ and $i=1,\dots ,7$. The metric supplies $G_{\mu \nu }$, seven vectors $G_{\mu i}$ and 28 scalar fields $G_{ij}=G_{ji}$. The three-form supplies a four-dimensional three-form $A_{\mu \nu \rho }$, seven two-forms $A_{\mu \nu i}$, $\left(\genfrac{}{}{0ex}{}{7}{2}\right)=21$ vectors $A_{\mu ij}$ and $\left(\genfrac{}{}{0ex}{}{7}{3}\right)=35$ scalars $A_{ijk}$.

A massless two-form in four dimensions has one physical particle polarization and can be dualized to a scalar by $H_3=\ast d\sigma$; the seven two-forms thus supply seven more scalars. The four-dimensional three-form has no local particle polarization, since its four-form field strength is locally a constant times the volume form. The total bosonic content is consequentlyThe eleven-dimensional Majorana spinor has 32 real components, decomposing into eight four-dimensional Majorana spinors. Accordingly ${\Psi }_{\mu }$ supplies eight gravitini, while ${\Psi }_i$ supplies $7\times 8=56$ spin-$1/2$ fields after diagonalizing the gravitino/matter kinetic terms. The resulting theory has 32 real supercharges, hence is four-dimensional N=8 supergravity, withThe on-shell counts check the reduction: the metric has $11(11-3)/2=44$ polarizations and the three-form has $\left(\genfrac{}{}{0ex}{}{9}{3}\right)=84$, totaling 128. In four dimensions $2+28\cdot 2+70=128$ bosonic polarizations match $8\cdot 2+56\cdot 2=128$ fermionic ones. Higher Kaluza–Klein levels, fluxes or projections would be additional choices beyond this massless toroidal reduction.

Interpret triviality as holomorphic triviality. Choose a global holomorphic bundle frame $s_1,\dots ,s_k$ for $E$. Let $H_{\alpha }$ have as its columns the coordinates of these sections in the prescribed local frame over $U_{\alpha }$, with the convention that a fiber coordinate vector transforms as $s_{\beta }=F_{\alpha \beta }{s}_{\alpha }$. ThenEach $H_{\alpha }$ is holomorphic and invertible because its columns form a basis in every fiber. This proves the requested holomorphic splitting of transition functions on a trivial bundle. Smooth or topological triviality alone would not justify holomorphic splitting.

For the line bundle on projective twistor space, pull its patching function back to the twistor line $L_x$ using the incidence convention of this problem, ${\omega }^A=x^{A{A}^{\prime }}{\pi }_{A^{\prime }}$, without an extra factor of $i$. Triviality on each line gives nonvanishing linewise holomorphic functions $H_0,H_1$ with $F_{01}=H_1{H}_0^{-1}$. The factors can be chosen as holomorphic functions of $x$ locally: a degree-zero scalar transition function has zero winding on the overlap annulus, hence admits a logarithm there; splitting its positive and negative Laurent series parts gives factors whose coefficients are Cauchy integrals that are holomorphic functions of $x$. The explicit splitting is worked out below. Their only common linewise ambiguity is multiplication by a nonzero function $g(x)$: the ratio is a global holomorphic function on the compact complex projective line, hence constant on that line.

Take ${ϵ}_{0^{\prime }{1}^{\prime }}=1$, ${ϵ}^{0^{\prime }{1}^{\prime }}=-1$ and raise a two-component spinor by left multiplication with ${ϵ}^{A^{\prime }{B}^{\prime }}$. The derivative along an alpha-plane is ${\ell }_A={\pi }^{A^{\prime }}{\partial }_{A{A}^{\prime }}$. It annihilates the unsplit twistor function, becauseThusis holomorphic on the whole line and has projective weight one. A section of $\mathcal{O}(1)$ is linear in the homogeneous coordinates: in an affine coordinate, holomorphic regularity at infinity bounds its growth to at most first order, so the Cauchy integral formula excludes all higher powers. ThereforeThe inverse splitting functions obey $({\ell }_A+G_A)H_{\alpha }^{-1}=0$. Commuting these two equations provesThe spinor decomposition of gauge curvature iswhere both $\phi$ two-component spinors are symmetric. Vanishing of the contraction for every $\pi$ forces ${\phi }_{A^{\prime }{B}^{\prime }}=0$. This is exactly the anti-self-dual Maxwell equations. Since the Abelian gauge curvature is $dA$, it is closed; its duality condition also makes it co-closed, giving the source-free Maxwell equations. Changing both splitting factors by $g(x)$ changes $A$ by $d\log g$, the expected Abelian gauge transformation. This derives the rank-one Penrose-Ward correspondence in the needed local form.

For the exponential patching function, choose a representative $f$ of the Čech cohomology class, of homogeneous degree zero. On the overlap of the two line patches write $f_x(\zeta )=f(x\pi ,\pi )={\sum }_{n\in \mathbb{Z}}{f}_n(x){\zeta }^n$. To compute the gauge potential explicitly, use a constant two-component spinor basis in which ${\iota }_{A^{\prime }}=(1,0)$ and the finite patch ${\pi }_{0^{\prime }}=\zeta ,{\pi }_{1^{\prime }}=1$. Then ${\pi }^{0^{\prime }}=-1$, ${\pi }^{1^{\prime }}=\zeta$ andFor a positively oriented contour in the overlap annulus, split its Laurent series asThe first is holomorphic inside the contour, the second outside including infinity; $H_{\alpha }=e^{h_{\alpha }}$ split $e^f$. Since $(-{\partial }_{B{0}^{\prime }}+\zeta {\partial }_{B{1}^{\prime }})f_x=0$, coefficient comparison gives ${\partial }_{B{0}^{\prime }}{f}_n={\partial }_{B{1}^{\prime }}{f}_{n-1}$. Applying that operator separately to $h_0,h_1$ cancels all nonconstant terms and leavesThus $-A_{B{0}^{\prime }}+\zeta A_{B{1}^{\prime }}=G_B$ yields $A_{B{1}^{\prime }}=0$ and $A_{B{0}^{\prime }}=-{\partial }_{B{1}^{\prime }}{f}_{-1}$. MoreoverSubstitution gives $A_{B{0}^{\prime }}=-(2\pi i{)}^{-1}\oint f_{,B}d\zeta$, which in homogeneous form is preciselyThis is the Abelian Ward splitting by Laurent series. The factors have weights $-1,-1,2$, respectively, so the integrand is invariant under a change of homogeneous representative. The contour avoids ${\iota }_{C^{\prime }}{\pi }^{C^{\prime }}=0$; the basis chosen above places that point at infinity. One can keep the contour fixed for $x$ in a sufficiently small domain, allowing differentiation under the integral.

Finally differentiate the contour potential for an anti-self-dual Maxwell field. By the twistor incidence relation, ${\partial }_{A{A}^{\prime }}{f}_{,B}={\pi }_{A^{\prime }}{f}_{,AB}$. ConsequentlyThe second equality uses the elementary two-spinor identity ${\pi }_{A^{\prime }}{\iota }_{B^{\prime }}-{\pi }_{B^{\prime }}{\iota }_{A^{\prime }}={ϵ}_{A^{\prime }{B}^{\prime }}({\iota }_{C^{\prime }}{\pi }^{C^{\prime }})$ in the declared convention. The remaining two-component spinor is symmetric in $A,B$, and no primed symmetric gauge curvature part survives. This directly verifies $dA$ is anti-self-dual. In complexified Lorentzian signature duality has Hodge star eigenvalues $\pm i$; here the vanishing primed two-component spinor fixes the stated ASD convention rather than incorrectly imposing the Euclidean equation $\ast F=-F$ on that signature.