By L. P. Hughston, R. S. Ward
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Extra info for Advances in Twistor Theory
Ays settle on a particular "sufficie ntly fine" covering ( U ) where . in effect. t here is no "relevant topology" lef t on i each U or intersections the r eof (i ,e, all the HP(U n " . n Uk ,G) vanish i i fOt' all p --' O- - although . as it stands . }(X . G), I shall henceforth /l ssume that 1 such a "suff i ciently" fine cove r'jng has been taken , ~nd that it is COuntable and locally finite. Now what does this definition have to do with the famniar "dual" relation to ordinary homology H (X .
Iv) V}I ... C' is positive frequency: a similar argument gives VAAl ~B' ... O' = O. Going the other way, let ~A' ... C be a spin-~n solution of the zeroI 4. rest-mass field equations on M+, depending holomorphically on the space. t es; 1et t AAeba constant, rea 1 tlme. 1 1. ke vector, norma l'lze d tlme coor d lna AA' = 2; and let B be the (O,I)-form on B+ defined by so that tAAlt I n + 1 -n-2 A C' -A B = 2TIl t ~. . t AA I lr I where t = tAAI -A TI TI AI • •• --c tee '/T I -f 11' dTI"E Then, by direct calculations, dB (12) 0 and lJ,JA' ••• CI (13) (where the left hand side can, as before, be interpreted as an integral over the complex projective line in projective twistor space corresponding to a fixed point in M+).
2 r ' A' .. • , K(X) (211 0 - ? c: x ""fA 2 ~ ! 'x ". I'" , 2 L' f (2 " ) d r. 2) (nm' under the Poincan~ t~listoY' space )101- gr oup , nor the lorentz gr oup) . (in There is a "gauge" freedom G \·/hereby fl>f-th -t-h+, where h' is hoIomorphic on some extended (iomain O~ ( ::> D) of twistor space in ~Ihich the contour 'I can be deformed to a pO i nt (to the "left" in V and to the "ri9ht" in V+ : see Fi9 , A) . (iii) G depends on the location of y (and ',- is not invar'iant under SU(2,2) -nor under the Poincar~ 9roup nor the lorentz qroup -- so G is r,ot invariant e i ther) , (iv) By inVOking G, then movi n9 I , then inVOking a ne~1 G, moving .
Advances in Twistor Theory by L. P. Hughston, R. S. Ward