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

We show that, in round spheres of dimension $n\ge 3$, for any given collection of codimension 2 smooth submanifolds $\mathfrak{S}:=\{{\mathrm{\Sigma }}_1,\mathrm{.}\mathrm{.}\mathrm{.},{\mathrm{\Sigma }}_N\}$ of arbitrarily complicated topology ($N$ being the complex dimension of the spinor bundle), there is always an eigenfunction $\psi =({\psi }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\psi }_N)$ of the Dirac operator such that each submanifold ${\mathrm{\Sigma }}_a$, 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.