The Penrose transform in split signature

A version of the Penrose transform is introduced in split signature. It relates cohomological data on CP3∖RP3CP3∖RP3 and the kernel of differential operators on M  , the (real) Grassmannian of 2-planes in R4R4. As an example we derive the following cohomological interpretation of the so-called X-ray transformHc1(CP3∖RP3,O(−2))→≅ker(□2,2:Γω(M,ε[−1]˜)→Γω(M,ε[−3]˜)) where Γω(M,ε[−1]˜) and Γω(M,ε[−3]˜) are real analytic sections of certain (homogeneous) line bundles on M, c   stands for cohomology with compact support and □2,2□2,2 is the ultrahyperbolic operator. Furthermore, this gives a cohomological realization of the so-called “minimal” representation of SL(4,R)SL(4,R). We also present the split Penrose transform in split instanton backgrounds.