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For affine complexified Minkowski space, the relevant twistor domain isIncidence can be written ${\omega }^A=x^{A{A}^{\prime }}{\pi }_{A^{\prime }}$, absorbing the conventional factor $i$ into the definition of $x$. Each space-time point gives a twistor line $L_x\cong {\mathbb{CP}}^1$. On charts ${\pi }_{0^{\prime }}\ne 0$ and ${\pi }_{1^{\prime }}\ne 0$, the fibre coordinates transform by $\omega /{\pi }_{1^{\prime }}=(\omega /{\pi }_{0^{\prime }})/\lambda$, with $\lambda ={\pi }_{1^{\prime }}/{\pi }_{0^{\prime }}$, giving the two $\mathcal{O}(1)$ summands. The full projective twistor space corresponds to complexified compactified Minkowski space.

Choose an antilinear quaternionic structure $J$ on ${\mathbb{C}}^4$. It induces a fixed-point-free involution on projective twistor space. A real Euclidean spacetime point is a $J$-invariant complex two-plane, equivalently a quaternionic line in ${\mathbb{H}}^2$. These form ${\mathbb{HP}}^1\cong S^4$, the conformal compactification of Euclidean space. Every projective twistor lies on the unique invariant line $\mathbb{P}{\mathrm{span}}_{\mathbb{C}}(Z,JZ)$, giving a fibration ${\mathbb{CP}}^3\to S^4$ with fiber ${\mathbb{CP}}^1$.

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.

Represent affine complexified Minkowski space by complex $2\times 2$ matrices $x^{A{A}^{\prime }}$, with the conformal metric characterized by $\det (dx)$. Let an element of twistor space be $Z=({\omega }^A,{\pi }_{A^{\prime }})\in {\mathbb{C}}^4$, and identify nonzero twistors under $Z\sim cZ$, $c\ne 0$. The compact projective twistor space is ${\mathbb{CP}}^3$. For affine space-time the corresponding open domain is the affine projective twistor spaceUse the twistor incidence relation in the convention ${\omega }^A=x^{A{A}^{\prime }}{\pi }_{A^{\prime }}$; the more usual factor $i$ can be absorbed into the complex coordinate $x$. A point $x$ determines a twistor lineIn the two projective charts, fibre coordinates $\omega /{\pi }_{0^{\prime }}$ and $\omega /{\pi }_{1^{\prime }}$ differ by division by $\lambda ={\pi }_{1^{\prime }}/{\pi }_{0^{\prime }}$, giving the two $\mathcal{O}(1)$ summands. Two distinct space-time points have intersecting lines precisely when $(x-y)\pi =0$ for some nonzero $\pi$, equivalently $\det (x-y)=0$; this recovers conformal null separation.

The twistor correspondence is the double fibrationFor a fixed twistor, the incidence solutions form an affine alpha-plane, a two-dimensional totally null plane. Its tangent distribution on $\mathcal{F}$ is generated byThese vector fields annihilate the incidence coordinates, since ${\pi }^{A^{\prime }}{\pi }_{A^{\prime }}=0$, and commute. This distribution is the geometric source of the linear system in the Penrose-Ward correspondence.

For the inverse construction, take a rank-$r$ holomorphic vector bundle $E$ on the twistor domain corresponding to a space-time open set $U$. The crucial hypothesis isThese are line-trivial bundles in the Ward correspondence. It is not enough to specify an arbitrary holomorphic vector bundle, nor just zero First Chern class on its lines: $\mathcal{O}(1)\oplus \mathcal{O}(-1)$ has degree zero but is nontrivial. A line with a nontrivial splitting is a jumping line and does not support the regular linewise construction there. Thus a globally nonsingular field on all affine complexified space-time requires triviality on all its relevant lines; a local construction uses only the lines over $U$.

Choose holomorphic vector bundle bundle frames on two twistor patches, and let $P$ be the invertible transition matrix on their overlap. Pull it back to $\mathcal{F}$. Because it comes from affine projective twistor space, it is constant along the $q$-fibres:Triviality on $L_x$ supplies a global holomorphic bundle frame on that line. Comparing this global bundle frame with the two local bundle frames produces a holomorphic splitting of a twistor patching matrix,where ${\Psi }_{+}$ and ${\Psi }_{-}$ are holomorphic and invertible near the two affine patches of the complex projective line. The splitting can be chosen holomorphically in $x$ locally: the restrictions have $r$ independent global sections and $H^1({\mathbb{CP}}^1,{\mathcal{O}}^{\oplus r})=0$, so these bundle frames extend through a neighborhood of the given line. This explains where regularity and linewise triviality enter the construction.

Differentiate the factorization using $V_{A}P=0$. Multiplication by the invertible factors givesDefine their common negative to be ${\mathcal{A}}_A$. It is regular on both projective patches and homogeneous of degree one in $\pi$, because $V_A$ has that degree while the bundle frames have degree zero. ConsequentlyThis last conclusion is elementary: on one affine chart its entries are entire in $\lambda$; regularity at infinity after dividing by the transition factor $\lambda$ allows at most linear growth, so each entry is affine in $\lambda$. Equivalently, a global section of $\mathcal{O}(1)$ is a linear homogeneous polynomial. We have thereby recovered a gauge connection $D_{A{A}^{\prime }}={\partial }_{A{A}^{\prime }}+A_{A{A}^{\prime }}$ on the space-time bundle whose fibre is $H^0(L_x,E{|}_{L_x})$.

The splitting factors now obeyTheir invertibility implies the compatibility conditionsIn the spinor decomposition of gauge curvature,the two spinors are symmetric. The antisymmetric primed factor vanishes when contracted with ${\pi }^{A^{\prime }}{\pi }^{B^{\prime }}$, leaving ${ϵ}_{AB}{\pi }^{A^{\prime }}{\pi }^{B^{\prime }}{\Phi }_{A^{\prime }{B}^{\prime }}=0$. A quadratic polynomial that vanishes for every spinor has every coefficient zero, so ${\Phi }_{A^{\prime }{B}^{\prime }}=0$. With the orientation convention used here, this is exactly anti-self-duality of gauge curvature, hence the ASDYM equations. This establishes the field equation from the holomorphic data, rather than assuming it as an extra input.

The construction naturally gives a gauge class. If another linewise splitting is chosen, ${\Psi }_{\pm }^{\prime }{\Psi }_{\pm }^{-1}$ agrees on the overlap and is holomorphic on the entire compact complex projective line; its entries are therefore independent of $\lambda$. Write this common matrix as $g(x{)}^{-1}$. Thenthe ordinary Yang-Mills gauge transformation. Changing the original holomorphic vector bundle bundle frames or replacing $E$ by an isomorphic bundle produces the same space-time gauge class. A determinant trivialization or another holomorphic structure-group reduction gives the corresponding restriction from $GL(r,\mathbb{C})$ to $SL(r,\mathbb{C})$ or another complex gauge group.

For perspective, the forward direction starts from an ASD gauge connection and uses its flat gauge covariant derivative along each alpha-plane to descend parallel sections to affine projective twistor space. The reconstructed linear system is exactly that parallel-section system. Going back through it recovers the original line-trivial bundle, while changing space-time bundle frames amounts to Yang-Mills gauge transformation. Thus, locally and with the stated regularity and triviality conditions, holomorphic line-trivial bundles determine solutions of the ASDYM equations uniquely up to gauge.

No unitary reality condition is imposed on complexified Minkowski space in this question. Obtaining real Euclidean gauge fields requires the compatible antiholomorphic twistor involution and a bundle reality structure; obtaining finite-action instantons requires further global and boundary conditions. Those are additional restrictions on the complex correspondence, not automatic consequences of having any holomorphic vector bundle on projective twistor space.

On complexified Minkowski spacetime, use the twistor incidence relationThe harmless factor $i$ is a conventional choice; omitting it corresponds to rescaling the complexified coordinates. For fixed $x$, projectivizing the nonzero two-component spinors $\pi$ gives a twistor line $L_x\simeq {\mathbb{CP}}^1$. The two stated patches cover all these incidence lines; they describe the affine-space-time twistor domain with the line $\pi =0$ removed from full projective twistor space.

Since $E{|}_{L_x}$ is holomorphically trivial, choose a global bundle frame on that line. Comparing its columns with the bundle frames in the two patch trivializations gives invertible matrices $H(x,\pi )$ and $\tilde{H}(x,\pi )$, holomorphic on their respective patches, withThis is the holomorphic splitting of a twistor patching matrix. It follows directly from existence of the global bundle frame, rather than from an arbitrary factorization of an arbitrary matrix: nontrivial line bundles would obstruct such a nonsingular splitting.

The factors can be chosen holomorphically in $x$ near any given $x_0$. An explicit local argument is useful. Normalize the known splitting at $x_0$ and set $Q(x,\lambda )=\tilde{H}(x_0,\lambda {)}^{-1}F(x,\lambda )H(x_0,\lambda )$ on a fixed circular contour in the overlap. It is close to the identity when $x$ is close to $x_0$. Let $P_{+}$ select strictly positive Laurent powers, and seek $q=1+$ positive powers satisfyingIn a holomorphic contour-function norm the operator has norm below one for sufficiently small variation, so its Neumann series converges and is holomorphic in $x$. Then $Qq=\tilde{q}$ has only nonpositive powers and is holomorphic on the exterior patch including infinity. Both factors are close to the identity and invertible. They extend over their full patches through $Q$ and its inverse. Taking $H=H(x_0)q$ and $\tilde{H}=\tilde{H}(x_0)\tilde{q}$ gives the required local family. This is also the elementary Laurent series reason that $H^1({\mathbb{CP}}^1,\mathcal{O})=0$ leaves no infinitesimal obstruction.

Raise primed two-component spinor indices with the antisymmetric two-component spinor form and define $V_A={\pi }^{A^{\prime }}{\partial }_{A{A}^{\prime }}$, holding $\pi$ fixed. Incidence givesThus $V_{A}F=0$ for a pulled-back function depending only on twistor coordinates. Differentiating the splitting and multiplying by the inverse factors givesThe left expression is holomorphic on one two-component spinor patch, the right on the other, and each is homogeneous of degree one in $\pi$. They therefore glue to a global matrix-valued section of $\mathcal{O}(1)$ on the line.

Such a section is linear in its homogeneous coordinates. To see this explicitly, set $\lambda ={\pi }_{0^{\prime }}/{\pi }_{1^{\prime }}$ on $U$. The normalized expression is an entire matrix function of $\lambda$; on the other patch its quotient by $\lambda$ is holomorphic at infinity. Hence it has at most linear growth. Cauchy estimates force all its derivatives of order at least two to vanish. Entry by entry it is $a_A(x)+\lambda b_A(x)$, or invariantlyThe coefficients are holomorphic in $x$. This proves existence of the desired space-time one-form, with no remaining dependence on $\pi$ in its coefficients.

A second splitting must differ by right multiplication $H\mapsto Hg(x)$ and $\tilde{H}\mapsto \tilde{H}g(x)$. Indeed the comparison matrix is holomorphic on both patches of a compact complex projective line and therefore constant along the line. The resulting transformation isthe gauge connection transformation law. Local coefficient matrices consequently glue to a holomorphic gauge connection on the bundle whose fiber at $x$ is the space of holomorphic sections of $E{|}_{L_x}$. If a single global matrix-valued one-form is wanted on all $M_{\mathbb{C}}\simeq {\mathbb{C}}^4$, choose a basis at the origin and parallel-transport it along the straight radial paths to each $x$. Linear transport exists on every finite path and varies holomorphically with $x$; this gives a global holomorphic bundle frame. The assertion does not require flat connection in all four directions, only a fixed choice of paths for a trivialization. On more general space-time domains the intrinsic object is a gauge connection with these gauge-related local one-forms.

Finally put $D_{A{A}^{\prime }}={\partial }_{A{A}^{\prime }}+{\Phi }_{A{A}^{\prime }}$ and ${\mathcal{D}}_A(\pi )={\pi }^{A^{\prime }}{D}_{A{A}^{\prime }}$. From the defining identity,Thus the columns of $H^{-1}$ are an invertible parallel bundle frame in the two alpha-plane directions. Their matrix commutator must annihilate that bundle frame, soUse the spinor decomposition of gauge curvaturewhere both gauge curvature two-component spinors are symmetric. The second term vanishes under contraction with ${\pi }^{A^{\prime }}{\pi }^{B^{\prime }}$. The first is a quadratic polynomial in the two two-component spinor components; vanishing for every two-component spinor forces each of its coefficients to vanish. Thereforewhich is precisely the ASDYM equation in the convention where the primed symmetric two-component spinor is the self-dual part. Equivalently, the gauge connection is flat when restricted to every alpha-plane. This completes the forward Penrose-Ward correspondence, including both the existence of the gauge connection and its anti-self-duality of gauge curvature.

In the compact twistor correspondence, a point $p$ of complexified compactified Minkowski space is a two-dimensional subspace $W_p\subset \mathbb{T}$. The correspondence fiber consists of flags $(\ell ,W_p)$ with $\ell$ a one-dimensional subspace of $W_p$. Its image in projective twistor space is precisely $\mathbb{P}(W_p)$, soThis is the twistor line of $p$. In the affine chart the twistor incidence relation gives $W_p=\{(i{p}^{A{A}^{\prime }}{\pi }_{A^{\prime }},{\pi }_{A^{\prime }}):\pi \in {\mathbb{C}}^2\}$. The linear map $\pi \mapsto (ip\pi ,\pi )$ is injective, and projectivizing it gives the same Complex projective space ${\mathbb{CP}}^1$. The subspace argument also covers points at conformal infinity, where that graph chart fails.

Set $d_0=0$ and assume $0<d_1<\cdots <d_m<n$. Choose the first member of the flag manifold as a $d_1$-plane, contributing $d_1(n-d_1)$ complex dimensions. After choosing $L_{i-1}$, choose $L_i/L_{i-1}$ as a $(d_i-d_{i-1})$-plane in $V/L_{i-1}$. Its Grassmannian has dimension $(d_i-d_{i-1})(n-d_i)$. Adding these fiber dimensions givesFor $V={\mathbb{C}}^4$ the results areHere $F_1={\mathbb{CP}}^3$ is projective twistor space, $F_2=\mathrm{Gr}(2,4)$ is complexified compactified Minkowski space, and $F_{12}$ is the twistor correspondence space of incident pairs consisting of a spacetime point and a twistor on its twistor line. Over $F_2$ its fiber is ${\mathbb{CP}}^1$; over $F_1$ its fiber is ${\mathbb{CP}}^2$, consistent with $4+1=3+2=5$. These are complex dimensions; the underlying real dimensions are $6,8,10$, while a real spacetime slice of $F_2$ has real dimension four.

Because $F_1$ is projective twistor space, the printed column represents the line $p=[e_1]$, not a distinguished nonzero vector. Its stabilizer subgroup consists of matrices whose first column is a nonzero multiple of $e_1$:The block form is necessary to preserve the line, and conversely every such determinant-one matrix preserves it. By transitivity, the orbit map $g\mapsto g[e_1]$ is onto $F_1$. Its two values agree precisely when $g^{-1}{g}^{\prime }\in H_p$, equivalently when $g{H}_p=g^{\prime }{H}_p$. This gives the homogeneous space identificationThe dimensions also agree: ${\dim }_{\mathbb{C}}SL(4)=15$, while the free $B$ and $b$ parameters give ${\dim }_{\mathbb{C}}{H}_p=9+3=12$; the quotient has dimension three.

This coset space is not a quotient group, because $H_p$ is not a normal subgroup. For example $h=I+E_{12}\in H_p$. Let $g$ have upper-left block $\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ and identity on the other two coordinates. Then $g\in SL(4,\mathbb{C})$ but $gh{g}^{-1}=I-E_{21}$ sends $e_1$ to $e_1-e_2$, so it is not in $H_p$. Thus multiplying coset representatives does not define a well-defined group operation.

If instead one required the literal vector $e_1$ to be fixed, one would have $a=1$ and $B\in SL(3,\mathbb{C})$; its quotient would be ${\mathbb{C}}^4\setminus \{0\}$, not $F_1$. The projective reading is required by the printed target quotient and the flag manifold definition.

The starting point is the Grassmannian of complex two-planes $W\subset \mathbb{T}\cong {\mathbb{C}}^4$. Choose a basis $u,v$ and form the simple bivector $P=u\wedge v\in {\Lambda }^2\mathbb{T}$. A change of basis multiplies $P$ by a nonzero determinant, so its projective class is independent of that choice. The Plücker embedding is thereforeIt is injective because $W=\{z:z\wedge P=0\}$. In a basis of $\mathbb{T}$, its six homogeneous coordinates $p_{ij}=u_i{v}_j-u_j{v}_i$ satisfyThis is the condition $P\wedge P=0$. Conversely, a nonzero antisymmetric $4\times 4$ matrix satisfying it has Pfaffian zero, hence rank at most two; its nonzero rank is even, so it has rank two and is a simple bivector. Therefore the equation characterizes the image, the Klein quadric $Q\subset {\mathbb{CP}}^5$. Its gradient cannot vanish at a projective point, since its six partial derivatives are the six coordinates up to permutation and sign. Thus $Q$ is smooth and has complex dimension four, agreeing with $\mathrm{Gr}(2,4)$.

The Klein quadric is the complexified compactified Minkowski space of the twistor correspondence. A volume form on $\mathbb{T}$ turns $P\wedge P^{\prime }$ into a scalar symmetric bilinear form $B(P,P^{\prime })$. It is zero exactly when the two underlying two-planes have nontrivial intersection: the four spanning vectors are then linearly dependent. In projective twistor space this says that the two twistor lines meet. In the affine graph chart supplied by the twistor incidence relation, it is $\det (x-y)=0$, so it is precisely null separation.

This also determines a conformal geometry on $Q$. At $[P]$, the tangent space is $P^{\perp }/⟨P⟩$ for $B$. The restriction of $B$ descends to a nondegenerate quadratic form on this four-dimensional quotient. Rescaling the representative $P$ changes the identification by a scalar, so its invariant content is a conformal class. In a graph chart its null tangent vectors are the rank-one matrices in $\mathrm{Hom}(W,\mathbb{T}/W)$, equivalently $d{x}^{A{A}^{\prime }}=a^A{b}^{A^{\prime }}$.

The compact complex geometry does not yet specify a real spacetime. The Lorentzian slice comes from the twistor Hermitian form $h$ of signature $(2,2)$: real points are maximal totally isotropic two-planes. In the incidence chart, isotropy of $W_x=\{(ix\pi ,\pi )\}$ is equivalent to $x=x^{†}$, the usual real Minkowski condition. Its twistor lines lie wholly in projective null twistor space. Diagonalizing $h$ as positive and negative two-dimensional blocks also gives a useful global description: an isotropic two-plane projects isomorphically to the positive block, so it is a graph $(z,Uz)$, and isotropy is $U^{†}U=1$. The real compact space is therefore $U(2)$, equivalently $(S^1\times S^3)/{\mathbb{Z}}_2$.

The same result is especially transparent in coordinates of the Plücker embedding after a complex linear change of coordinates on ${\Lambda }^2\mathbb{T}$. The real Lorentzian cone has equationIts nonzero real generators represent points. Normalize the lengths of its positive and negative parts to one; real projectivization identifies their simultaneous antipodes. ConsequentlyThis is the Projective Minkowski compactification. Its universal cover is $\mathbb{R}\times S^3$, with deck generator $(s,v)\mapsto (s+\pi ,-v)$, so its fundamental group is $\mathbb{Z}$. The projective version includes periodic identifications; an unwrapped Einstein-cylinder or Penrose diagram description uses the appropriate cover and affine domain instead.

For example, use homogeneous cone coordinates $[U,V,t,\mathbf{x}]$ with quadratic equation $2UV+t^2-|\mathbf{x}{|}^2=0$. The affine patch $U\ne 0$ sets $U=1$ and then $V=-(t^2-|\mathbf{x}{|}^2)/2$. The complement $U=0$ is the null cone $t^2-|\mathbf{x}{|}^2=0$ at infinity. It is much larger than a single point: different null directions retain different limits there.

For the Euclidean slice, choose an antilinear quaternionic structure on a complex vector space $J$ on twistor space, for exampleThe Euclidean reality structure on twistor space selects $J$-invariant complex two-planes. Such a plane is a quaternionic line in ${\mathbb{H}}^2$, givingThere are no fixed projective twistors for $J$: if $JZ=cZ$, applying $J$ again would give $-Z=|c{|}^{2}Z$. Every projective twistor instead lies on the unique invariant line $\mathbb{P}\mathrm{span}(Z,JZ)$. Thus the Euclidean correspondence becomes a fibration ${\mathbb{CP}}^3\to S^4$ with fiber ${\mathbb{CP}}^1$, in contrast to the Lorentzian correspondence where only null twistors represent real null geodesics. Distinct Euclidean real points have disjoint twistor lines, since the line through any $Z$ and $JZ$ is unique. This reflects the absence of nonzero real null vectors for a positive-definite metric.

On the Klein quadric the Euclidean real cone has signature $(1,5)$; normalize its timelike coordinate to one and obtain the unit sphere $S^4$. Its affine Euclidean chart omits one point. Inverse stereographic projection explicitly gives the round conformal metricwhich extends smoothly across that point. This proves the one-point conformal compactification of Euclidean space. The distinction between the Lorentzian Hermitian and Euclidean quaternionic reality structures is also described in Adamo's lectures on twistor theory; the global statements here follow from the explicit models just derived.

Finally, Wick rotation $t\mapsto it$ compares affine real slices through their complexification. Locally it changes $d{t}^2-|d\mathbf{x}{|}^2$ to $-d{\tau }^2-|d\mathbf{x}{|}^2$, the negative of the Euclidean metric; the overall sign does not change its null cone in the complexified setting. It does not identify the two real conformal compactifications globally. The Lorentzian projective space has fundamental group $\mathbb{Z}$, while $S^4$ is simply connected. Its causal universal cover $\mathbb{R}\times S^3$ is noncompact, so passing to that cover does not repair the mismatch with the compact Euclidean sphere.

There is also a direct algebraic obstruction. In the homogeneous coordinates above the affine Wick substitution extends only as the rational mapSubstitution verifies that the image satisfies the same complex quadric equation. For $U=0$ and $t\ne 0$ it collapses the point to $[0,1,0,0]$, while at some infinity points every output coordinate vanishes. In particular it is indeterminate at $[0,1,0,0]$. This is a genuine failure of holomorphic extension: the affine paths $(t,\mathbf{x})=(0,(s,0,0))$ and $(s,(is,0,0))$, as $s\to \infty$, both tend to that input point. Their Wick images tend respectively to $[0,1,0,0]$ and $[0,0,1,(1,0,0)]$, which are distinct points of the complex quadric.

Thus the answer to the global compatibility question is no: ordinary affine Wick rotation is a local analytic continuation, not a globally regular invertible map of the conformally compactified real spacetimes. The global obstruction to Wick rotation of conformal compactifications does not prevent both real slices from being studied inside the same smooth complex Klein quadric.

The flag manifold of a line $\ell$ contained in a two-plane $W$ has projections $(\ell ,W)\mapsto W$ and $(\ell ,W)\mapsto \ell$. Their fibers are respectively ${\mathbb{CP}}^1$ and ${\mathbb{CP}}^2$. This is the flat twistor correspondence: complexified compactified Minkowski space is the Grassmannian of two-planes in twistor space, while projective twistor space is its projective space of lines.

In flat twistor theory, a twistor is a pair $Z=({\omega }^A,{\pi }_{A^{\prime }})$ of two-component spinors. The twistor incidence relation with a spacetime point is ${\omega }^A=i{x}^{A{A}^{\prime }}{\pi }_{A^{\prime }}$. The nonzero vector $Z$ and its projective class play different roles: the latter belongs to projective twistor space.

Twistor theory encodes the conformal geometry of four-dimensional spacetime using incidence geometry in a complex vector space. In the flat correspondence, a spacetime point is a two-plane in a four-dimensional twistor space, and hence a projective line in projective twistor space. Null separation becomes intersection of these lines. Choosing a spacetime signature is an additional reality condition on the complex geometry.