The Penrose transform in split signature

Abstract

A version of the Penrose transform is introduced in the split signature. It relates the cohomological data on CP 3 \ RP 3 and kernel of differential operators on M , the (real) Grassmannian of 2-planes in R 4 . As an example we derive the following cohomological interpretation of the so-called X-ray transformwhere Γ ω (M, ε[−1]) and Γ ω (M, ε[−3]) are real analytic sections of certain (homogeneous) line bundles on M , c stands for cohomology with compact support and 2,2 is the ultrahyperbolic operator. Furthermore, this gives a cohomological realization of the so-called "minimal" representation of SL(4, R). We also present the split Penrose transform in split instanton backgrounds.

Self-dual Yang–Mills equations in split signature

“…Let ∇(A, B) be the projection of the trivial connection on C 2k+2n onto E(A, B). Computations similar to the above proposition shows that the curvature of ∇ is of type (1,1). Therefore there is a holomorphic structure on E(A, B) defined by ∇ 0,1 .…”

“…quaternions x such that x 2 = xx = 1. Then its Lie algebra, sp (1), is identified with the purely imaginary quaternions, i.e. quaternions with…”

“…This transform is known as the X-ray transform [1,16]. One can Identify Γ(RP 3 , ε(−2)) with homogeneous functions on R 4 − {0} of degree −2.…”

“…1 is again an ASDYM solution. It is known that the proper conformal group of H R P 1 is isomorphic to SL(4, R)/{±1}.…”