|WikiProject Mathematics||(Rated Start-class, Low-priority)|
I'm thinking of breaking off spinc structure into a separate article. After all, it's longer than the section on the spin structure itself, which isn't so strange as it's more complicated/interesting (although there's a lot that one could add to the spin structure section about applications various theories (and exotic stuff like supersymmetry without supersymmetry) and one could talk about various properties of spin manifolds, like the canonical division by two of the first Pontrjagin class which gives Witten's characteristic class, etc ... although I don't know if I know enough about any of these topics to include them myself). Any comments/objections? JarahE 19:41, 24 April 2006 (UTC)
- I also think it could use breaking off. The section on spin^c structures should probably be considered a fragment and in need of development. How vague can you get? "A spinc structure is analogous, but uses the spinc group, which is the U(1)-extension of SO(n) built on top of the extension by the cyclic group of order 2 in Spin(n)." What analogy are you using? Which U(1) extension are you talking about? etc. It's too vague. And there are no references. None of the references in this article mention spin^c structures. Rybu
- Well, I found references at least for the definition of spin^c structures. I put in a proper definition (which should really be polished) and mentioned some of the most basic facts. I appended the Gompf reference. It's certainly not the original reference for the concept, but it was one of the easiest ones to find on-line. Who originated the notion of spin^c ? Rybu —Preceding signed but undated comment was added at 04:05, 12 October 2007 (UTC)
This seemed out of place
I removed the following paragraph, since it certainly didn't belong in the intro, and it may not even belong in the article at all:
- In particle physics the spin statistics theorem implies that the wavefunction of an uncharged fermion is a section an associated vector bundle to a the spin lift of an SO(N) bundle E. Therefore the choice of spin structure is part of the data needed to define the wavefunction, and often needs to be summed over in the partition function. In many physical theories E is the tangent bundle, but for the fermions on the worldvolumes of D-branes in string theory it is the normal bundle.
Silly rabbit 19:55, 21 June 2006 (UTC)
- Spin structures are important in particle physics. Particle physics in turn was crucial to the history of spin structures, the name term "spin structure" itself I think owes its origins to particle physics. So the connection between the two I think needs to appear in the article, if not in the introduction, although I agree that the above paragraph is too technical for an introduction. Perhaps a compromise would be a nontechnical sentence in the introduction and then the above paragraph can be put in a particle physics subsection analogous to that in the spin^c section. --JarahE 07:35, 2 July 2006 (UTC)
- Fair enough. Shall we create a Spin structures in particle physics section (or such-like), and point it out in the intro? Silly rabbit 19:12, 2 July 2006 (UTC)
- I vote to keep Spin-c here. Morally, these are just another "kind" of "spin" structures (a propos of spinors). Silly rabbit 21:36, 2 July 2006 (UTC)
- Ok, so then maybe we should put in a redirect so that people searching for spin^c structures get here? I'm not sure how to do this with the superscript, maybe redirect all of the various ways that people could try to type it? JarahE 12:02, 3 July 2006 (UTC)
We have some problems developing in the spin^c part of this article. I think the problem is Sławomir Biały is editing an article that has a less general definition of spin^c structure than he would like. Currently the definition is only for orientable bundles but he seems to be editing for a more general notion of spin^c structure. IMO there's no point in putting in results about a notion of spin^c structure that hasn't been defined in the article. Rybu (talk) 17:51, 20 May 2009 (UTC)
- Yes, you're right. I was confused. The Gompf article doesn't appear to give details on the correspondence. The Lawson and Michelsohn text gives a construction in which spinc structures naturally correspond to elements of . It could be that when the obstruction vanishes, this can be put into correspondence with ; however, I believe it has been said somewhere that the correspondence of spinc structures with is not natural. Sławomir Biały (talk) 21:46, 20 May 2009 (UTC)
Spin structures (starting with principal bundles)
You are starting the article Spin structure considering spin structures on vector bundles, but a (classical) spin structure on an oriented Riemannian manifold M is a spin structure on its tangent bundle TM (this is also the definition of spin manifold, see Lawson and Michelson (1989) "Spin Geometry" [page 78 (Spin Structures on Vector Bundles) and page 85 (Spin Manifolds and Spin Cobordism)]. I think is much better to start with a "classical definition" of spin structure, using principal fibre bundle, cf. Thomas Friedrich in "Dirac Operators in Riemannian Geometry" (2000), page 35. This is also the original definition given by A. Haefliger, Sur l’extension du groupe structural d’un espace fibré, C. R. Acad. Sci. Paris 243 (1956) 558–560. For spin^c structure cf. (again!) Thomas Friedrich page 47. Mgvongoeden 16:52, 8 June 2011 (UTC)
- Lawson, H. Blaine; Michelsohn, Marie-Louise (1989). Spin Geometry. Princeton University Press. ISBN 978-0-691-08542-5.
- Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN 978-0-8218-2055-1
This article is a total train wreck.
It would be far better to remove this article entirely than to allow the world to see just how bad a Wikipedia article can be.
Better yet, of course, would be to rewrite the first part of the article, at least through the definition of the article's subject. Currently it is virtually incomprehensible, thanks to a vast absence of coherent writing. Can someone expert in the subject matter and in exposition please rewrite at least the first few sentences, the "Introduction", and the definitions of "spin structures", "spin structures on vector bundles", and "spinC structures? That could make this article worth retaining.Daqu (talk) 14:44, 11 December 2012 (UTC)
- The rest of the article is in desperate need of a rewrite as well. It is mentioned that the spinc group [sic!] is "defined" by the exact sequence
0 -> Z2 -> Spinc -> SO(n) x U(1) -> 1,
- which it most certainly is not.
- The group U(1) is mentioned numerous times before someone bothers to explain that it is the same as SO(2) or the circle group.
- Many items are repeated in different contexts as though mentioned for the first time, which is confusing. For example, under Obstruction (before getting to spinc structures) the fact that the second Stiefel-Whitney class is the obstruction to a spin structure is mentioned -- in about four different ways that all say the same thing, as though written by four different people -- all in the same brief paragraph.
- Under Details, an exact sequence of groups is used to get an exact cohomology sequence, which is then ignored in the text that immediately follows.
- Under the section Obstruction (after getting to spinc structures), the obstruction is defined in terms of "the bundle", although spinc structures are said to be defined on manifolds, not particularly on bundles. (Whereas spin structures are defined for bundles, but this is not mentioned for spinc structures.)
The comment(s) below were originally left at several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section., and are posted here for posterity. Following
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|An expert needs to look at this (or someone needs to copy something better out of a book):
One needs to distinguish a spin structure on a manifold from the spin-c group itself (Riemannian manifolds may or may not admit spin structures, not "spin groups"---one is lifting the structure group of the (principal frame bundle of the) manifold from so(n) to spin(n))
Defn of spin-c is not clear---how is it defined by exact sequence? (one can alway take spin(n)\times U(1), so presumably there should be some condition that the sequence does not split?)
Also one should note the compact group Spin(n) has a complexification (aften denoted Spin_n(C)) which is a complex reductive group, double covering the complex reductive group SO_n(C)) This is different to spin-c which is still compact (according to the definition here).
The wiki article on spin strucutres actually defines spin-c (so maybe this page should just be deleted?)220.127.116.11 (talk) 22:14, 4 September 2008 (UTC)
Substituted at 21:59, 26 June 2016 (UTC)