Geometric structures in the nodal sets of eigenfunctions of the Dirac operator

Abstract

We show that, in round spheres of dimension $n\geq3$, for any given collection of codimension 2 smooth submanifolds $\mathfrak{S}:=\{\Sigma_1,...,\Sigma_N\}$ of arbitrarily complicated topology ($N$ being the complex dimension of the spinor bundle), there is always an eigenfunction $\psi=(\psi_1,...,\psi_N)$ of the Dirac operator such that each submanifold $\Sigma_a$, modulo ambient diffeomorphism, is a structurally stable nodal set of the spinor component $\psi_a$. The result holds for any choice of trivialization of the spinor bundle. The emergence of these structures takes place at small scales and sufficiently high energies.