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We all know that the existence of spinor fields implies that spacetime must be time-orientable. Thus that spacetime is time-orientable is a necessary condition for existence of spinor fields.

Geroch, R. (1968). Spinor structure of space-times in general relativity I, J. Math. Phys. 9, 1739-1744 proved this theorem: In a non-compact spacetime the existence of 4 continuous vector fields constituting a Minkowski tetrad at each point is necessary and suffienct for the existence of spinor fields.

My qustions:

  1. For compact spacetimes, are there some necessary and suffienct conditions for existence of spinor fields?

  2. For general $n$-dimensional Lorentzian manifolds, what's the necessary and suffienct conditions for existence of spinor fields?

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Advanced Classical Field Theory (2009) by Giachetta, Mangiarotti, Sardanashvily remarks on p. 248:

  • A non-compact world manifold admits a Dirac spinor structure if and only if it is parallelizable.
  • For a compact world manifold $X$, its Euler characteristic and the second Stiefel-Whitney class $w_2$ must be zero, and its first Pontryagin number must be multiple of 48.

and gives the references

Geroch, R. (1968). Spinor structure of space-time in general relativity, J. Math. Phys. 9, 1739.

Wiston, G. (1974). Topics on space-time topology, Int. J. Theor. Phys. 11, 341.

World manifolds are assumed orientable, simply connected and 4-dimensional, so you need to look at the references to see if it applies to arbitrary dimensions.

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