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## What is the relation between photons and the eigenspinors of the Dirac operator?

Suppose we have shown with clear evidence that the universe is compact and four dimensional (excluding time).  One question would be whether it is spin.  Suppose we find that it is spin.  At this point the machinery examined by Atiyah-Singer apply and we have a rich structure: spinor bundles, spin connection, Dirac operators etc. I have been attempting to show that it is a perfect scaled 4-sphere which happens to be spin.  In that case, Christian Bar has shown that the spectrum of the Dirac operator are all integer multiples of $h$.  Thus it is an extremely natural question whether photons can be related to these eigenspinors of the Dirac operator and possibly we can understand the geometry of photons.