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As you more than probably know spinors dimensions go as $2^{\frac{D}2}$ in D spacetime dimensions. If we look at the peculiar case of D=2*4, spinors have 4 components and we usually say that's related to the 4 "degrees of freedom": particle/antiparticule & spin up/down?*. So basically linked to the degeneracy in energy and spin. But if we know go to D=6, they have 8 components, I am asking myself if it's not the appearance of a new physical degeneracy, but I don't really see what could it be. Maybe the group of rotation in 6 dimension has one more Casimir operator. Am I seeing it in a totally wrong way?

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This is a property of the compatibility of Weyl and Majorana conditions, and this happens periodically in dimension with period 4. Weyl fermions have half the dimension of Dirac fermions, Majorana fermions half again, so that when Weyl and Majorana conditions are compatible, you get 1/4 the number of components.

The Weyl-Majorana dimensions are 2,6,10, and you are noticing the issue with 6. The two and ten dimensional cases are equally interesting, being the worldsheet and spacetime dimensions of string theory with a free worldsheet action.

This is reviewed well in many string theory sources.

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