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.

main page
– abbreviations
– journals – comments
– other sites – acknowledgements

send feedback and suggestions to bombelli at olemiss.edu – modified 11 dec 2016