Energy quantization for Dirac systems over non-collapsed degenerating Einstein manifolds

We study energy quantization for a class of Dirac systems on compact spin Einstein manifolds of dimension $n$. For a sequence of solutions to a nonlinear Dirac system with uniformly bounded energy on a fixed spin Riemannian manifold, we first establish an energy identity theorem. We then investigate the more complicated case of underlying domain manifolds being a sequence of non-collapsed degenerating spin Einstein manifolds. At an orbifold singular point, three types of bubble spinors can possibly appear, living respectively on ${\mathbb{R}}^n$, on a Ricci-flat ALE bubble space, and on the flat cone ${\mathbb{R}}^n\mathrm{/}\mathrm{\Gamma }$. By developing asymptotic analysis for solutions over degenerating neck regions, we establish that energy identity holds.