Underlying Mathematical Structures
This chapter begins with a review of four definitions of twistors in curved space-time proposed by Penrose in the seventies, i.e. local twistors, global null twistors, hypersurface twistors and asymptotic twistors. The Penrose transform for gravitation is then re-analyzed, with emphasis on the double-fibration picture. Double fibrations are also used to introduce the ambitwistor correspondence, and the Radon transform in complex analysis is mentioned. Attention is then focused on the Ward picture of massless fields as bundles, which has motivated the recent analysis by Penrose of secondary potentials which supplement the Rarita-Schwinger potentials in curved space-time (see chapter eight). The boundary conditions studied in chapters seven and eight have been recently applied in the quantization program of field theories. Hence the chapter ends with a review of progress made in studying bosonic fields subject to boundary conditions respecting BRST invariance and local supersymmetry. Interestingly, it remains to be seen whether the Atiyah-Patodi-Singer theory of Riemannian four-manifolds can be applied to obtain an explicit proof of gauge invariance of quantum amplitudes.
KeywordsMathematical Structure Twistor Space Null Geodesic Spinor Field Conformal Curvature
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