🌟 Twistor Bundle Demo

Bundles over PT = CP³ of rank k map to U(1), SU(2), SU(3) curvatures via Penrose-Ward correspondence. This interactive demo shows how different bundle ranks generate the Standard Model gauge sectors through holomorphic geometry. [RFT 13.5 §F]

Because PT is complex projective three-space, its holomorphic frame bundle has SU(4) ≅ Spin(6) symmetry; this global symmetry acts on the twistor tetrad but does not replace the separate gauge bundles built on top (see equations below).

📐 Penrose-Ward Correspondence: Bundle → Gauge

Note: The arrows (→) represent the Penrose-Ward transform, which converts twistor cohomology classes into spacetime gauge fields. This is the mathematical bridge between the abstract bundle geometry and physical forces.

Rank-1 Bundle → U(1) Hypercharge
\\[ H^1(PT, \\mathcal{O}(-2)) \\rightarrow F_{\\mu\\nu}^{U(1)} \\]
Line bundle over projective twistor space generates electromagnetic field tensor
Rank-2 Bundle → SU(2) Weak Force
\\[ H^1(PT, E^{(2)}) \\rightarrow W_{\\mu\\nu}^a \\]
Rank-2 vector bundle generates SU(2) weak gauge field with 3 components (a = 1,2,3)
Rank-3 Bundle → SU(3) Strong Force
\\[ H^1(PT, E^{(3)}, c_2 = 3) \\rightarrow G_{\\mu\\nu}^A \\]
Rank-3 bundle with Chern class c₂ = 3 generates SU(3) gluon field with 8 components (A = 1...8)

🔍 Mini-Proof: Rank-3 Bundle → Three Chiral Families

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📐 Example: Rank-1 Bundle → Maxwell's Equations

\\[ H^1(PT, \\mathcal{O}(-2)) \\xrightarrow{\\text{PW}} \\partial_{[\\mu}A_{\\nu]} = F_{\\mu\\nu}, \\quad \\partial^{\\mu}F_{\\mu\\nu} = 0 \\]

A harmonic (0,1)-form in \\(\\mathcal{O}(-2)\\) maps to a free U(1) field; imposing self-duality picks out the on-shell Maxwell solutions. The Penrose-Ward transform explicitly converts twistor cohomology classes into spacetime field equations.

🎮 Interactive Bundle Explorer

Select a bundle rank to see its corresponding gauge sector and physical properties:

📊 Twistor Space Bundle Map

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📚 Mathematical Reference

Complete field equations and derivations

🧮 Equation Explorer

Interactive mathematical framework

🔬 Mathematical Methods

Advanced techniques and theory comparison

🌌 Theory Overview

Conceptual foundations of RFT