Analytic definition of spin structure
  We work on a parallelizable time-orientable Lorentzian 4-manifold and prove that in this case the notion of spin structure can be equivalently defined in a purely analytic fashion. Our analytic definition relies on the use of the concept of a non-degenerate two-by-two formally self-adjoint first order linear differential operator and gauge transformations of such operators. We also give an analytic definition of spin structure for the 3-dimensional Riemannian case. 
Zhirayr Avetisyan, Yan-Long Fang, Nikolai Saveliev, Dmitri Vassiliev
math.DG gr-qc math-ph math.MP