Spin Structures on Manifolds| 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.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 13 feb 2021