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 T*M* 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 π_{1}(*B*)
= \(\mathbb Z\)_{2} × π_{1}(*M*),
which, for *M* simply connected, reduces to π_{1}(*B*)
= \(\mathbb Z\)_{2} (the only other possibility
besides the trivial group for such an *M*).

- __Nasc__: *w*_{2}(*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 π_{1}(*M*),
so they are unique iff *M* is simply connected.

* __Majorana spinors__: The
equation of motion for a Majorana spinor field is ∇^{A}_{B} *ψ*^{B} ≡ ∇_{a} *γ*^{aA}_{B} *ψ*^{B} =
0.

**Special Types of Manifolds and Topics**

* __3D__: All 3-manifolds
have a spin structure; S^{3} has only 1, but
with *n* handles it has 2^{n}.

* __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 Spin_{C}(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 spin^{c}-manifolds]; Finster & Kamran 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 a1611 [analytic definition].

@ __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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