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T h i s space i s a f i b r e bundle over S4 w i t h f i b r e S2. he map i s given by (z1,z2,z3,z4) + z1 + ~ 2 j / z 3 + z 4 j . - s4 (zl + z2j/z3 + z 4 j ) 47 ON YANG-MILLS FIELDS The SU(N) bundle P over S4 p u l l s up t o a bundle over CP3, and t h e f i b r e can be complexified t o SL(N,c) [see Eq. (3211. That i s . the SWN) f i b r e over a D o i n t D o f S4 i s l i f t e d t o a l l t h e S2 p o i n t s i n CP3 over p. The-group SU(N) i s enlakged t o t h e complex special 1inear group SL(N,C), constructing a new p r i n c i p a l buedle Pc over CP3.

236-275. , Topological and canpactness methods f o r some nonlinear v a r i a - J . Funct. Anal. 41 (1981) t i o n a l problems o f Mathematical Physics, t o appear. L. , boundary problems. I n Free boundary problems, Pavia, 1979 (Roma, 1980). , Topological methods i n t h e c a l - culus o f v a r i a t i o n s (Hermann, Paris, 1934). , On a non l i n e a r d i f f e r e n t i a l equation a r i s i n g i n nuclear phy- s i c s , Proc. Roy. I r i s h Acad. 62 (1963) 117-135. H. , Variational methods f o r nonlinear eigenvalue problems.

And (*F)14 = F23. Also l e t * denote d u a l i t y o f two forms i n So (*F)12 = F34, (*F)13 = -Fe4 D u a l i t y decomposes t h e f i e l d F i n t o i t s s e l f dual F+ and a n t i - s e l f dual Fcomponents; these components are the (+1)and (-1) eigenvalues under *. Ordinary two forms--not j u s t two forms w i t h values i n SU(N)--split t h e same way. Two forms a r e denoted by A 2 , the space o f skew symmetric 2-tensors. The s p l i t t i n g gives where A2 are the self-dual and a n t i - s e l f - d u a l o r d i n a r y forms respectively.

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