Metrics with harmonic spinors on even dimensional spheres (Q2739724)

The Atiyah-Singer index theorem implies a topological lower bound on the dimension of the space of harmonic spinors: NEWLINE\[NEWLINE \dim\ker(D) \geq |\widehat{A}(M)|,NEWLINE\]NEWLINE where \(D\) is the Dirac operator on a compact Riemannian spin manifold \(M\) without boundary. It has been conjectured that the dimension of the space of harmonic spinors cannot be bounded from above by a (differential) topological invariant when \(\dim(M) \geq 3\). In dimension 2 special phenomena occur. Note that the situation differs considerably from the case of harmonic differential \(p\)-forms where the dimension is itself topological (the \(p\)-th Betti number). Evidence for this conjecture is given by the fact that harmonic spinors are not topologically obstructed if \(\dim(M) \equiv 0,1,7 \bmod 8\) as shown by \textit{N. J. Hitchin} [Adv. Math 14, 1-55 (1974; Zbl 0284.58016)] or if \(\dim(M) \equiv 3 \bmod 4\) as shown by \textit{C. Bär} [Geom. Funct. Analysis 5, 899-942 (1996; Zbl 0867.53037)]. This means that in those dimensions every compact spin manifold has a Riemannian metric with nontrivial harmonic spinors. NEWLINENEWLINENEWLINEMore evidence comes from examples. The 1-parameter family of Berger metrics on \(S^n\), \(n \equiv 3 \bmod 4\), yields for special values of the parameter metrics with arbitrarily large kernel of the Dirac operator. On the other hand, it was not even known whether or not \(S^4\) has a Riemannian metric with nontrivial harmonic spinors. NEWLINENEWLINENEWLINEIn the present work the author shows that given any \(k\) and any \(n\) divisible by 4 there is a Riemannian metric on \(S^n\) such that \(\dim\ker(D) \geq k\). The metrics constructed in the proof have a large symmetry group; \(\text{Sp}(n/4)\) acts isometrically with principal orbits of codimension 1. It is also shown that such a construction would not be possible with \(\text{Sp}(n/4)\) replaced by \(\text{U}(n/2)\) or \(\text{SO}(n)\). The proof strongly uses methods from harmonic analysis, representation theory, and some asymptotic properties of ordinary differential equations.