The analysis of the massless Dirac equation on a manifold without boundary involves the topological concept of spin structure. We show that in lower dimensions spin structure can be 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. Our analytic construction works in dimensions four (Lorentzian signature) and three (Riemannian signature). We prove that our analytic definition of spin structure is equivalent to the traditional topological definition. A detailed exposition is provided in arXiv:1611.08297.
Spin structure has implications for spectral analysis. Say, a 3-torus has eight different spin structures and, hence, eight different massless Dirac operators. These eight different massless Dirac operators have different spectra.