APureAxialSpecializationoftheTorsionfulTwistor Correspondence

This record contains the preprint and source files for: Hiroyuki Shioiri,A Pure Axial Specialization of the Torsionful Twistor Correspondence (2026). This work studies a pure axial specialization of the torsionful twistor correspondence on a four-dimensional Lorentzian spin manifold with a metric-compatible connection with torsion. The central idea is to separate null geometry from transport: the metric determines the null and spinorial structure, while the torsionful connection governs the transport law. Starting from Esposito’s torsionful α-surface formalism, the paper restricts torsion to the totally antisymmetric sector and shows that the relevant torsional data collapse to the axial one-form SAA'. In this sector, the contracted integrability condition reduces to a transport equation for the projected torsion component φ_A = πA'SAA', driven by a corresponding projection of the anti-self-dual Weyl spinor. In this precise transport-theoretic sense, the usual curvature obstruction to α-surface integrability is reorganized as a balance between Weyl curvature and torsional transport. :contentReference[oaicite:0]{index=0} To make the mechanism explicit, the manuscript formulates the theory in a principal-spinor-adapted dyad and studies a minimal vacuum plane-wave model. In the exact Petrov type N case, the principal transport equation becomes homogeneous; a small aligned Weyl perturbation then drives φ_A through an explicitly integrable equation. The analysis also shows that principal transport does not determine the full axial torsion one-form: in the adapted dyad, the decomposition SAA' = φ_A ιA' + χ_A oA' separates the transport-visible component φ_A from a complementary component χ_A, which remains free unless additional boundary, symmetry, or geometric conditions are imposed. The paper is intended as a structural first step toward a broader torsionful twistor geometry. Version 2 extends the original manuscript by adding a fixed-dyad two-projection reconstruction of the axial torsion one-form. Besides the transport-visible component φ_A, the complementary projection φ'_A is introduced as additional initial/boundary data, yielding the pointwise reconstruction SAA'=φ_AιA'-φ'_A oA'. The revised version also makes explicit the local recovery formula Ψ_1=(2i)⁻¹dΦ/du and clarifies the motivation for the minimal closure ∇^c_ φ'_A=0. Version 3 extends the earlier analysis by introducing the second projection φ'A := ιA' S{AA'} as complementary initial data. Together with the first transport-generated projection φ_A, the two-projection formula SAA' = φ_A ιA' − φ'A o{A'} yields an explicit assembly of the full axial torsion one-form from one dynamical component and one propagated complementary datum. This reconstruction is deliberately one-directional: the first projection is obtained from the principal residual transport equation, while the second is supplied as initial data on a null slice and extended by a minimal closure condition ∇_ℓ φ'_A = 0. A companion paper develops a genuinely two-direction reconstruction in which both projections are obtained dynamically from residual transport equations in a dual aligned Weyl sector (doi:10.5281/zenodo.19022720). Files included:- shioiri_pure_axial_twistor.pdf — main manuscript- README.md — repository and citation information Keywords:Twistor correspondence; α-surfaces; pure axial torsion; Riemann–Cartan geometry; spinor geometry; Weyl curvature; plane-wave spacetimes; integrability. :contentReference[oaicite:2]{index=2}