It is said that if we take the spacetime manifold to be a sphere $S^d$ of large volume so that the spectrum of Dirac operator $$i\gamma^\mu D_{\mu}$$ is discrete.

For example at least for $d=4$, this has been discussed in p.2 of Witten paper from this Physics Letters B, Volume 117, Issue 5, 18 November 1982, Pages 324-328 Physics Letters B, 117(5), 324–328

My question is simple that

1. why the spectrum of Dirac operator $i\gamma^\mu D_{\mu}$ is discrete? in any $d$? Can the spectrum of Dirac operator be continuum?

In what setup can it be discrete?

In what setup can it be continuum?

I thought that in a finite volume, the eigenvalues may have discrete. In an infinite volume, the eigenvalues may have a chance to have a part to be continuum?

2. under what conditions do we have zero eigenvalues for Dirac operator $i\gamma^\mu D_{\mu}$?

p.s. I know this is related to the Atiyah Singer index theorem. But I am trying to ask for an explicit user-friendly down-to-earth answer. I am not asking for a reference only. (The paper I quoted is a reference itself.)