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?

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.)

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.)