Ivancevic_Applied-Diff-Geom

698 Applied Differential Geometry: A Modern IntroductionFinally, we observe that via this machinery any spinorial quantity maybe identified with a (weighted) twistor. For example, valence 1 spinors inE A′ [w 1 ] or E A [w 2 ] may be dealt with via (4.173) or (4.174) respectively.This determines an identification for tensor powers by treating each factorin this way. This does all cases since, via (4.166),p−1{ }} {E A ∼ = E|B · · · D|[1], E A ′∼ = E|B ′ · · · C ′ | [−1].} {{ }q−1Now, any irreducible representation of G 0 is given as a tensor product oftwo irreducible components in tensor products of the fundamental spinors(viewed as representations of the special linear groups, adjusted by aweight). Applying the corresponding Young symmetrizers [Penrose andRindler (1984); Fulton and Harris (1991)] to the tensor products of E α andE β , we get the explicit realization of each irreducible spinor bundle as thesubbundle of the (weighted) twistor bundle which is isomorphic to the injectingpart of the twistor bundle. Thus a section of a weighted irreduciblespinor bundle V may be identified with a twistor object which is zero in allits composition factors except the first. So, in fact, this non–zero factor isalso the projecting part of the twistor. We write Ṽ for this twistor (sub–)bundle satisfying V ∼ = Ṽ. Altogether, we have established the followingresult [Gover and Slovak (1999)]: Any irreducible spinor object v can beidentified with the twistor ṽ which has the spinor as its projecting part.This identification is provided in a canonical algebraic way.4.13.3.2 Twistor CalculusGiven a choice of scale ξ, a twistor connection ∇ a on E α and E α is givenby [Penrose and MacCallum (1972); Dighton (1974); Bailey and Eastwood(1991)]and∇ P ′A( ) (vB=v B′∇ P ′A v B + δ B Av P ′∇ P ′A v B′ − P P ′ B ′AB vB )(4.176)∇ P ′A (u B :: u B ′) = (∇ P ′A u B + P P ′ B ′AB u B ′ ::: ∇ P ′A u B ′ − δ P ′B ′u A). (4.177)Notice that whereas on the left hand side ∇ indicates the twistor connection,on the right hand side the symbol ∇ indicates the usual spinor connection