Science > Physics > Number of distinct spin structures - is this true?
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Science > Physics |
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"" |
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04 Jun 2007 06:21:55 PM |
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Number of distinct spin structures - is this true? |
Hi all, I have a conjecture about the number of distinct spin bundles
corresponding to a given SO(n,m) bundle, which I guess is either well
known or false, and would appreciate it if somebody can tell me which.
First some definitions:
Suppose given a vector bundle E over a manifold M with structure group
SO(n,m). Let {Ui} be an open covering of M such that E is trivial over
each Ui, and let the transition functions corresponding to some set of
local trivialisations be tij. There is a homomorphism phi: Spin(n,m) --
SO(n,m) which is two-to-one, and which has the property that phi(A)
=3D phi(-A) (I don't know whether this is true in general but it is in
all the cases of interest to me). A spin bundle associated to E is a
bundle over M with structure group Spin(n,m) such that there exist
local trivialisations over Ui whose transition functions sij satisfy
phi(sij) =3D tij.
My conjecture is this: assuming E admits at least one spin structure,
the number of distinct (up to projection-preserving homeomorphism)
spin bundles associated to E is equal to the size of the =C4=8Cech
cohomology group H^1 ( {Ui}; F), where F is the sheaf which assigns
the multiplicative group {1,-1} to each connected open set.
If anyone would like to see the reasoning (which relies on a step
which seems reasonable to believe but which I can't prove) that led to
the conjecture I will be happy to provide it.
-Rotwang
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| User: "Bob Cain" |
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| Title: Re: Number of distinct spin structures - is this true? |
06 Jun 2007 02:39:58 AM |
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wrote:
Hi all, I have a conjecture about the number of distinct spin bundles
corresponding to a given SO(n,m) bundle, which I guess is either well
known or false, and would appreciate it if somebody can tell me which.
Rotwang, I'd suggest submitting this to sci.physics.research. Looks
like something John Baez might like to chew on (or chew up as the case
may be.)
Bob
--
"Things should be described as simply as possible, but no simpler."
A. Einstein
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| User: "" |
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| Title: Re: Number of distinct spin structures - is this true? |
06 Jun 2007 01:12:28 PM |
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On 6 Jun, 08:39, Bob Cain <arc...@arcanemethods.com> wrote:
s...@hotmail.co.uk wrote:
Hi all, I have a conjecture about the number of distinct spin bundles
corresponding to a given SO(n,m) bundle, which I guess is either well
known or false, and would appreciate it if somebody can tell me which.
Rotwang, I'd suggest submitting this to sci.physics.research. Looks
like something John Baez might like to chew on (or chew up as the case
may be.)
Bob
Thanks for the advice, I will give it a try.
-Rotwang
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| User: "Thomas Mautsch" |
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| Title: Re: Number of distinct spin structures - is this true? |
06 Jun 2007 06:04:41 PM |
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In news:<1180999315.663339.8630@g4g2000hsf.googlegroups.com>
schrieb <>:
Hi all, I have a conjecture about the number of distinct spin bundles
corresponding to a given SO(n,m) bundle, which I guess is either well
known or false, and would appreciate it if somebody can tell me which.
It is false by the way your definition of spin structure is stated;
but it becomes true and well-known if you define (as usually done)
a spin structure
as a pair
of a bundle with structure group Spin(n,m) *together with*
a 2-to-1 covering map from this bundle to the SO(n,m) principal bundle
of the SO(n,m) structure you start with;
furthermore this covering map has to be non-trivial on the fibers.
First some definitions:
Suppose given a vector bundle E over a manifold M with structure group
SO(n,m). Let {Ui} be an open covering of M such that E is trivial over
each Ui, and let the transition functions corresponding to some set of
local trivialisations be tij. There is a homomorphism
phi: Spin(n,m) --> SO(n,m) which is two-to-one,
and which has the property that phi(A) = phi(-A)
(I don't know whether this is true in general but it is in
all the cases of interest to me). A spin bundle associated to E is a
bundle over M with structure group Spin(n,m) such that there exist
local trivialisations over Ui whose transition functions sij satisfy
phi(sij) = tij.
My conjecture is this: assuming E admits at least one spin structure,
the number of distinct (up to projection-preserving homeomorphism)
spin bundles associated to E is equal to the size of the ?ech
cohomology group H^1 ( {Ui}; F), where F is the sheaf which assigns
the multiplicative group {1,-1} to each connected open set.
For spin structures on manifolds with positive-definite metric,
you can find the correct statement with definitions and proof in
H.B.Lawson Jr., M.-L.Michelsohn
"Spin Geometry", Princeton Univ. Press, 1989,
Part II, Chapter 1.
The proof carries over to general semi-Riemannian metrics
(as far as I can see and do remember...).
Note, however, that Lawson and Michelsohn's
classifiation of spin^c structures in one of the appendices
of the book is wrong. - The book is known for containing some
serious mistakes; but it is still one of the best
references on spin geometry I have seen so far.
The fact that a bundle with structure group Spin
might give rise to different spin structures
is due to John Milnor, and is also discussed in the book.
I would think that Milnor's paper
"Spin structures on manifolds"
L' Enseignement Math 9 (1963) 198-203
also contains the classification of spin structures. -
It is one of the first mathematical papers on spin structures,
if not *the* very first one.
Concerning more recent references:
The classification of spin structures
is certainly also contained in the more recent book
by Thomas Friedrich on "Dirac Operators in Riemannian Geometry"
(available both, in German and in English).
Probably also in many recent introductions in Seiberg-Witten theory;
although Seiberg-Witten theory is mainly about spin^c structures...
In fact more than what you conjectured is true:
Although the correspondence between the Cech cohomology
you mention above - of the manifold with values in Z_2 -
and spin structures is not a canonical one,
the Cech cohomology classes *act* *in a canonical way*
freely and transitively on the space of spin structures.
-
A set with such a free and transitive action of a group G
is nowadays called a G-torsor.
If you are really interested, I might be able to track
some further references for you.
John Baez has a webpage about torsors.
Hope this helps!
Thomas Mautsch
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