Open AccessSolutions to Yang—Mills equations that are not self-dual
Author(s) -
L. M. Sibner,
Robert J. Sibner,
Karen Uhlenbeck
Publication year - 1989
Publication title -
proceedings of the national academy of sciences of the united states of america
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 5.011
H-Index - 771
eISSN - 1091-6490
pISSN - 0027-8424
DOI - 10.1073/pnas.86.22.8610
Subject(s) - instanton , invariant (physics) , mathematical physics , mathematics , action (physics) , yang–mills theory , dual (grammatical number) , yang–mills existence and mass gap , magnetic monopole , partial differential equation , gauge theory , differential equation , physics , mathematical analysis , quantum mechanics , art , literature
The Yang—Mills functional for connections on principleSU (2) bundles overS 4 is studied. Critical points of the functional satisfy a system of second-order partial differential equations, the Yang—Mills equations. If, in particular, the critical point is a minimum, it satisfies a first-order system, the self-dual or anti-self-dual equations. Here, we exhibit an infinite number of finite-action nonminimal unstable critical points. They are obtained by constructing a topologically nontrivial loop of connections to which min—max theory is applied. The construction exploits the fundamental relationship between certain invariant instantons onS 4 and magnetic monopoles onH 3 . This result settles a question in gauge field theory that has been open for many years.