Path-connected Yang-Mills moduli spaces
1984; Lehigh University; Volume: 19; Issue: 2 Linguagem: Inglês
10.4310/jdg/1214438683
ISSN1945-743X
Autores Tópico(s)Algebraic Geometry and Number Theory
ResumoMin-max techniques in the calculus of variations are used to prove that the moduli spaces of self-dual connections on principal SU(2) or SU(3) bundles over S 4 are path-connected. IntroductionOn a principal bundle P -> S 4 whose structure group, G, is a compact, simple and simply connected Lie group, there are distinguished connections.These are the connections whose curvature is self-dual with respect to the Hodge dual of the metric on T*S 4 which is induced from the identification S 4 = {x e R 5 : \x\ 2 = 1}.(This metric is called the standard metric.)The moduli space of self-dual connections on P, 3Jt(P) = (P s x (smooth, self-dual connections on P})/Aut P, is a smooth manifold.Here P s is the fibre of P at s = south pole, and Aut P is the group of smooth automorphisms of P. The isomorphism class of P is specified by its integer degree, k(P) [4].(For G = SU(2), k(P) = -c 2 (P X SU(2) C 2 ).)If £(/>)< 0, then Wl(P)= 0; if k(P) = 0, then 2W(P) = point; and if k(P) > 0, then Wl(P) is nontrivial.Although these spaces have been the subject of much recent study, [4], [11], [14], relatively little is known of their global structure.A small advance is made in this article with the following theorem.Theorem 1.1.Let P -> S 4 be a principal G = SU(2) or SU(3) bundle with positive degree.Then Wl(P) is path-connected.
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