The Penrose Transform in the Split Signature

TL;DR

This paper introduces a version of the Penrose transform in split signature, linking cohomological data with differential operators and representations on real Grassmannians, with applications to X-ray transforms and instanton backgrounds.

Contribution

It develops a novel split signature Penrose transform connecting cohomology, differential operators, and representations, expanding the transform's applicability.

Findings

Cohomological interpretation of the X-ray transform

Cohomological realization of the minimal representation of SL(4,R)

Extension of the Penrose transform to split instanton backgrounds

Abstract

A version of the Penrose transform is introduced in the split signature. It relates the cohomological data with supports on the open subsets of the complex 3-projective space and kernel of differential operators on the (real) Grassmannian of 2-planes in the Euclidean 4-space. As an example we derive a cohomological interpretation of the so-called X-ray transform. Furthermore, a cohomological realization of the so-called "minimal" representation of SL(4,R) is given. We also present the split Penrose transform in split instanton backgrounds.