A Morse theory for equivariant Yang-Mills
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 action, are classified by a pair of positive, odd, integers \(m,n\) which give the weights of the action of the stabiliser over the exceptional orbits. By calculating the index of the relevant deformation operator, regarded as a virtual representation of \(SU(2)\), and applying vanishing theorems, the author shows that there are no instanton solutions on these equivariant bundles when \(m,n>1\). Thus the absolute minimum of the Yang-Mills functional on the invariant connections on any equivariant bundle with \(m,n>1\) gives the desired Yang-Mills solution. The other main result involves Yang-Mills-Higgs fields: these are solutions of a coupled system of equations involving a pair of a connection and a Higgs field -- a section of an associated bundle -- and they are the critical points of the Yang-Mills-Higgs functional. The author gives the first proof of the existence of nontrivial solutions of the Yang-Mills-Higgs equations over \(S^ 4\), using a similar method. These solutions minimise the Yang-Mills-Higgs functional among the invariant pairs, but not over the space of all pairs, in general. In the appendix to the paper, the author extends the partial compactness results for the Yang-Mills functional, due to Uhlenbeck, Sedlacek and Taubes, to the Yang-Mills-Higgs case.
scientific article; zbMATH DE number 78542
scientific article