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

TL;DR

This paper demonstrates that on high-dimensional spheres, one can construct Dirac eigenfunctions with nodal sets matching any prescribed complex topology of codimension 2 submanifolds, revealing intricate geometric structures at small scales.

Contribution

It proves the existence of Dirac eigenfunctions with prescribed complex topological nodal sets in high-dimensional spheres, a novel geometric realization result.

Findings

Existence of eigenfunctions with prescribed nodal set topology

Nodal structures appear at small scales and high energies

Results hold for any trivialization of the spinor bundle

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.