Existence of Dirac eigenvalues of higher multiplicity

Abstract

In this article, we prove that on any compact spin manifold of dimension m ≡ 0, 6, 7 mod 8, there exists a metric, for which the associated Dirac operator has at least one eigenvalue of multiplicity at least two. We prove this by "catching" the desired metric in a subspace of Riemannian metrics with a loop that is not homotopically trivial. We show how this can be done on the sphere with a loop of metrics induced by a family of rotations. Finally, we transport this loop to an arbitrary manifold (of suitable dimension) by extending some known results about surgery theory on spin manifolds.

A universal spinor bundle and the Einstein–Dirac–Maxwell equation as a variational theory

“…[AWW12; AWW14]. One can also construct the identification isomorphisms β g,h without the partial connection, see [Mai97], and show that these give the union L 2 (ΣM ) := g∈Sm,0(M ) L 2 (Σ g M ) the structure of a Hilbert bundle over the Riemannian metrics, see [Now15,Chapter 4]. But since the base and the fiber of that bundle are both infinite dimensional, the space of sections of this bundle does not have a canonical Fréchet space topology, which makes it difficult to do calculus in this space.…”

An Obstruction to Higher-Dimensional Kernel of Dirac Operators

“…, ±l k } and every eigenvalue is simple. Nikolai Nowaczyk proved in [22] that on a manifold with dimension 0, 6, 7 mod 8 there exist a metric such that the Dirac operator has at least one eigenvalue of multiplicity at least two.…”