Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 6914.13.3 Penrose Twistor CalculusRecall that the twistor theory, originally developed by Roger Penrose in1967, is the mathematical theory which maps the geometric objects of the4D Minkowski space–time into the geometric objects in the 4D complexspace with the metric signature (2, 2). The coordinates in such a space arecalled twistors.The twistor approach appears to be especially natural for solving theequations of motion of massless fields of arbitrary spin.Recently, Ed Witten used twistor theory to understand certain Yang–Mills amplitudes, by relating them to a certain string theory, the topologicalB model, embedded in twistor space. This field has come to be known astwistor string theory.4.13.3.1 Penrose Index FormalismExcept where otherwise indicated we use Penrose’s abstract index notation[Penrose and Rindler (1984)] which allows for easy explicit calculationswithout involving a choice of basis. Thus we may write, v A or v B for asection of the unprimed fundamental spinor bundle E A . Similarly w A ′ coulddenote a section of the primed fundamental spinor bundle E A ′. We write E Afor the dual bundle to E A and E A′ for the dual to E A ′. The tensor productsof these bundles yield the general spinor objects such as E AB := E A ⊗ E B ,EA ABC′ ′ B and so forth. The tensorial indices are also abstract indices. Recall′that E a = EA A is the tangent bundle, so E ′ a = EAA′ is the cotangent bundleand we may use the terms ‘spinor’ or ‘section of a spinor bundle’ to describetensor fields [Gover and Slovak (1999)].A spinor object on which some indices have been contracted will betermed a spinor contraction (of the underlying spinor). For example,vBC ABC′ ′ DE is a contraction of vABC′ DD ′ EF . In many cases the underlying spinorof interest is a tensor product of lower valence spinors. For example,v AB wB C′ u ACD is a contraction of v AB wC C′ u DEF . The same conventionsare used for the tensor indices and the twistor indices; the latter are tointroduced below. Standard notation is also used for the symmetrizationsand antisymmetrizations over some indices.Weights and scalesWe define line bundles of densities or weighted functions as follows [Goverand Slovak (1999)]. The weight -1 line bundle E[−1] over M is identified