Non-self-dual Yang-Mills connections with nonzero Chern number. (English) Zbl 0722.58014
We prove the existence of non-self-dual Yang-Mills connections on SU(2) bundles over the four-sphere with standard Riemannian metric. In particular, our proof covers all bundles with second Chern number \(C_ 2\neq \pm 1.\) Existence on the trivial bundle \(C_ 2=0\) was previously established by L. M. Sibner, R. J. Sibner, and K. Uhlenbeck [Proc. Natl. Acad. Sci. USA 86, No.22, 8610-8613 (1989)] via different methods. We consider connections that are invariant under an SU(2) symmetry group acting by bundle morphisms. The Yang-Mills equations reduce to a system of ODE’s with singularities. Existence of invariant Yang-Mills connections is proved via variational methods. The nonexistence of invariant (anti-)self-dual connections is proved by studying the boundary value problem for the associated first-order ODE system. The explicit form of the non-self-dual connections is not known.
Reviewer: J.Segert
MSC:
| 58E15 | Variational problems concerning extremal problems in several variables; Yang-Mills functionals |
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |
Keywords:
nonzero Chern number; non-self-dual Yang-Mills connections; invariant Yang-Mills connections; variational methodsReferences:
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