Spin structures and spinor bundles

For an oriented Riemannian rank-n real vector bundle with n at least two, a spin structure is a principal Spin(n) bundle together with an equivariant double covering of its SO(n) frame bundle. Local transition functions can each be lifted, but their products on triple overlaps may acquire signs. The resulting obstruction is the second Stiefel–Whitney class w₂. The oriented bundle admits a spin structure exactly when this class vanishes in degree-two cohomology with Z/2 coefficients.

When a lift exists, its isomorphism classes form a torsor for H¹(M;Z/2). A torsor has a simply transitive action, rather than a preferred zero element: calling one structure trivial requires an additional reference choice. A connected simply connected base therefore has a unique spin structure when it has any at all. Orientation is a separate prerequisite, expressed by w₁=0; a nonorientable manifold does not become spin merely by making an isolated statement about w₂.

An associated complex spin representation turns the chosen principal spin bundle into the spinor bundle S. In dimension 2m its complex rank is 2^m, split equally between the two chiralities. The Levi-Civita connection lifts to this bundle and produces the untwisted spin Dirac operator. Distinct spin structures may change global holonomy and zero modes even with the same metric and local differential expression. Index invariance under changes of connection within a fixed construction does not identify all these spectra.

Spin-c geometry is a related but different lift, using (Spin(n)×U(1))/{±1}. Its determinant line has first Chern class reducing modulo two to w₂; the corresponding existence obstruction is the integral Bockstein W₃. Every almost-complex manifold has a natural spin-c structure, but it need not be spin. Spin-c Dirac index densities contain an additional exponential of half the determinant first Chern class. That factor cannot be silently dropped in favor of the untwisted spin formula.

For complex projective space the tangent Euler sequence computes the total Chern class as (1+H)^(n+1), truncated above its dimension, with H the positive hyperplane class. Thus the spin obstruction is the reduction of (n+1)H. This gives simple oriented manifolds where a spin assumption is valid or false. One must check the relevant cohomology class, rather than assuming that complex geometry supplies an ordinary spin structure.

On a flat torus a chosen global tangent framing makes the spin structures describable by periodic or antiperiodic signs around the generating circles. They shift the allowed Fourier frequencies by zero or one half. A nonzero shift can remove every parallel spinor without affecting ellipticity or the numerical index. These examples separate three layers: the existence of a lift, the choice among possible lifts, and the analytic kernel associated with the resulting connection and metric.