A Morse theory for equivariant Yang-Mills
type: Item
instance of: scholarly article
title: A Morse theory for equivariant Yang-Mills
zbMATH Open document ID: 0782.58016
DOI: 10.1215/S0012-7094-92-06610-5
author: Thomas H. Parker
published in: Duke Mathematical Journal
publication date: 1992-12-14T00:00:00Z
review text: The Yang-Mills equations are Euler-Lagrange equations for the Yang-Mills functional: the \(L^ 2\) norm of the curvature tensor of a connection. The ``Condition C'' of Palais and Smale gives a criterion under which variational methods such as Morse theory can be applied in infinite dimensional problems, but in the Yang-Mills case the condition fails to hold in the crucial 4-dimensional case. This failure stems from the breakdown of the Sobolev inequalities in the case of the critical exponent. Thus it is not at all a straightforward matter to find solutions using variational methods, although a great deal of work has been done in this direction. In the first part of this paper the author shows that one can recover Condition C if one restricts attention to connections over a Riemannian 4-manifold \(M\) which are invariant under the action of a group \(H\), acting by isometries on \(M\) with all orbits of strictly positive dimension. To show this, he develops a variant of the Sobolev theory for \(H\)-invariant functions and shows that for such functions one can improve the exponents to those which hold for functions in dimension \(4-d\), where \(d\) is the smallest dimension of an \(H\)- orbit. In the second part of the paper the theory is applied to obtain specific existence results in the case when \(M\) is the 4-sphere. There are well- known instanton solutions (self-dual or anti-self-dual) to the Yang-Mills equations, and the first main result is the existence of other, non- instanton, solutions. (Such solutions were found slightly earlier by Uhlenbeck, Sibner and Sibner, using another variational approach, and by Sadun and Segert, using ODE methods.) To find these, the author considers the action of \(SU(2)\) on \(S^ 4\) derived from the irreducible representation of \(SU(2)\) on \(\mathbb{R}^ 5\). There are two exceptional orbits, copies of the real projective plane, and the other orbits are 3-dimensional. The equivariant \(SU(2)\) bundles over \(S^ 4\), for this…
Mathematics Subject Classification ID: 58E15
Mathematics Subject Classification ID: 81T13
zbMATH DE Number: 78542
zbMATH Keywords: Palais-Smale condition C
zbMATH Keywords: Yang-Mills-Higgs fields
zbMATH Keywords: Yang-Mills functional
zbMATH Keywords: vriational methods
reviewed by: S. K. Donaldson
MaRDI profile type: Publication
cites work: Self-duality in four-dimensional Riemannian geometry
cites work: The index of elliptic operators. III
cites work: Chern numbers, quaternions, and Berry's phases in Fermi systems
cites work: Q5658847
cites work: Stability and isolation phenomena for Yang-Mills fields
cites work: The Smith Conjecture in dimension four and equivariant gauge theory
cites work: Pseudofree orbifolds
cites work: Q5185713
cites work: Morse theory on Hilbert manifolds
cites work: Lusternik-Schnirelman theory on Banach manifolds
cites work: Q5548503
cites work: The principle of symmetric criticality
cites work: Gauge theories on four dimensional Riemannian manifolds
cites work: Conformal fields and stability
cites work: Non-minimal Yang-Mills fields and dynamics
cites work: A direct method for minimizing the Yang-Mills functional over 4- manifolds
cites work: Non-self-dual Yang-Mills connections with quadrupole symmetry
cites work: Solutions to Yang—Mills equations that are not self-dual
cites work: Path-connected Yang-Mills moduli spaces
cites work: A framework for Morse theory for the Yang-Mills functional
cites work: Connections with \(L^ p \)bounds on curvature
description: scientific article; zbMATH DE number 78542