METRICS, CONNECTIONS AND GLUING THEOREMS 11 denote a complex vector bundle over X wit h fiber C2 and with projectio n TT:V—X. a) Th e anti-sel f dua l equation s Recall fro m th e first lectur e tha t th e anti-sel f dua l equation s fo r a covariant derivativ e V o n V (or , equivalently , fo r a connectio n o n V ) form a se t o f algebrai c equation s fo r th e curvature , R v . Choos e a n oriented, orthonorma l basi s {e 1 ,... , e4} for T*X nea r som e point, an d expand R v wit h respec t t o th e induce d base s fo r A 2 T*X a s i? v = 2 _ 1 E 4 J=1 i?^e2 A eK Then , th e anti-sel f dualit y equation s becom e th e simple assertio n tha t (2.1) R 12 + R 3 4 = 0 , R 2 3 + R 14 = 0, R 31 + R 2 4 = 0 . Of course, these equations should best b e thought o f as a set of equa- tions fo r th e covarian t derivative . T o put thi s las t statemen t int o som e sort o f usefu l form , conside r (2.1 ) o n a n ope n se t U C X wher e ther e exists a vecto r bundl e isomorphis m g : V\ v « U x C 2 . Here , on e ca n compare th e covarian t derivativ e V with g~ x d- g, wher e d is the covari - ant derivativ e on the bundle UxC 2 whic h sends a section s (a C 2 -valued function o n U) t o ds, wher e d i s the usua l exterio r derivative . A s wa s explained i n the first lecture , gVg~~ l d + a, wher e a i s a 1-for m o n U with value s i n u(C 2 ). Becaus e d 2 = 0 , the curvatur e R v i s given b y (2.2) g R v g~l = da + a A a, where th e secon d ter m involve s exterio r produc t o f form s an d matri x multiplication. Wit h the preceeding understood i t follow s that (2,1 ) on U i s equivalent t o th e followin g first orde r differentia l equatio n fo r a : (2.3) P+(da + aAa) = 0 where P + : A2T*X A + i s the (fiberwise ) orthogona l projection . Unless V i s isomorphic t o I x C 2 , the n (2.3 ) i s at best , a n equatio n which is only locally defined. I f you are uncomfortable wit h this, one can preceed a s follows: First , choos e a favorite metri c compatibl e covarian t derivative, Vo , o n V. Introduc e th e u(V r )-vaiued 1-form , a = V V o and i? v i s then give n b y (2.4) R v = R Vo + d Vo a + a A a. Here, d Vo i s the covarian t exterio r derivativ e fo r V o this being define d as follows: Us e the Levi-Civit a covarian t derivativ e o n section s o f T*X
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