Spin Structures on Manifolds  

In General > s.a. Soldering Form; stiefel-whitney classes.
* Idea: One introduces a soldering form for each tangent space, so that σ is now a field; Spinors (with a sign forgotten) can be interpreted as null flags, null vectors with half-planes attached; These are close enough to geometrical objects to allow definition of covariant derivatives.
$ Spin structure: For an oriented 4-manifold M, a spin structure is a lift of the structure group of TM from SO(4) to its double cover Spin(4) = SU(2) × SU(2).
- Necessary condition for existence: M must be orientable and time-orientable.
- Nasc: The principal fiber bundle B of oriented, time-oriented bases on M, is not simply connected, and can be written as π1(B) = \(\mathbb Z\)2 × π1(M), which, for M simply connected, reduces to π1(B) = \(\mathbb Z\)2 (the only other possibility besides the trivial group for such an M).
- Nasc: w2(M) = 0, but this is not a useful condition in practice.
- Nasc: If M is non-compact, it must be parallelizable (this requires that the null flag bundle possess the appropriate double covering).
* Classification: Spin structures are classified by π1(M), so they are unique iff M is simply connected.
* Majorana spinors: The equation of motion for a Majorana spinor field is ∇AB ψB ≡ ∇a γaAB ψB = 0.

Special Types of Manifolds and Topics
* 3D: All 3-manifolds have a spin structure; S3 has only 1, but with n handles it has 2n.
* 4D: The spin group can be written Spin(4) = SU(2) × SU(2), and each element represented as a 4 × 4 block matrix with two SU(2) matrices on the diagonal, and is contained in the 7D Lie group SpinC(4).
* Extension: Every compact, oriented 4D manifold possesses (at least one) Spin\(_C\) structure.
@ 2D: Szabados CQG(08)-a0712 [and quasilocal quantities in general relativity].
@ 3D: Deloup & Massuyeau Top(05) [space of complex spin structures].
@ Non time-orientable spacetime: Friedman CQG(95); Chamblin & Gibbons CQG(95)gq.
@ In curved spacetime: Fatibene et al GRG(98)gq/96; Lisi gq/98; Peeters & Waldron JHEP(99)ht [with boundary, index theorems]; Fatibene & Francaviglia IJGMP(05) [and Ashtekar variables]; Ikemakhen JGP(06) [pseudo-Riemannian spinc-manifolds]; Finster & Kamran MAMS(19)-a1403 [on singular spaces].
@ Generalizations: Cimasoni LMP(09)-a0804 [discrete, from oriented graphs].

References > s.a. differential geometry; killing felds; Pin Structures.
@ General: Geroch JMP(68), JMP(70) [in general relativity]; Bugajska IJTP(79) [Crumeyrolle's approach]; Sardanashvily JMP(98)gq/97 [covariant]; Trautman AIP(98)ht; Morrison BS(01)mp; Avetisyan et al JMP(17)-a1611 [analytic definition]; Murcia & Shahbazi a2102 [globally hyperbolic 4-manifolds].
@ Lie and covariant derivatives: Hurley & Vandyck JPA(94), JPA(94), JPA(95); > s.a. lie derivatives.
@ Related topics: Barrett MPCPS(99)gq/95 [and skein spaces]; Schray et al JMP(96)gq [and degenerate metrics].
> Online resources: see Wikipedia page.

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