Discussion:
Isometries and fibre bundle, kk reduction question
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gmoutso
2005-04-05 17:14:23 UTC
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Dear users,
I'm trying to understand dimensional reduction of a field theory (eg.
sugra) on a coset space from a geometric perspective, ie I don't care
yet for the field eom.

My question is this, given a group of isometries G on a manifold E
(that preserves metric and the other fields), we can construct on
E the space of orbits under G {Op}. The group G doesn't necessarily
act transitevely.

Is the space {Op} a manifold? Are there any conditions for it to be
a manifold? If it is then we must have a fibre bundle E->M={Op}
where the projection takes each point p on E into its orbit
[p]=Op={q:q=gp,g in G}.

If the action og G is free then the orbit Op should be diffeomorphic to
G for each p. Would that correspond to a group reduction (in the
literature group reductions, torus reductions and coset reductions are
considered seperate)

For those interested I'll move on. The above question must be related
to foliations and G structures (I don't know much about them) but if that
is true then my guess is that E is foliated into orbits (locally?) if at
each p in E the number of nonzero independant killing vectors is constant.

Please take all of these statements as questions :)
The fibre Op should be isomorphic to G/H (H<G) but I can't see why H
should be the same for each Op. Anyway, given that Op=G/H for each p we
have a G/H fibre bundle E over M={[p]=Op, [p]=[q] if p=gq}. So far what
conditions I've used or could use would further make this bundle trivial
(globally) E=M x (G/H)?

What is similar or different to coset reductions eg. sphere reductions?
Are the problems in coset reductions related just to the consistency of
the field equations?

If the bundle construction is true, then I suppose an ansatz for the
metric and other fields is easy. eg. The metric on E would be a twisted
metric

g=g_M + exp(phi) g_{G/H} using some connection. I think I understand
how to find a metric on a homogenous space G/H where G acts as an
isometry and transitively.

Comments? Corrections?
Is there an elegant article on this topic using differential geometry?

Thanks,
George Moutsopoulos
Edinburgh
Urs Schreiber
2005-04-05 18:51:04 UTC
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"gmoutso" <***@yahoo.co.uk> schrieb im Newsbeitrag
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> I'm trying to understand dimensional reduction of a field theory (eg.
> sugra) on a coset space from a geometric perspective
[...]
> Is the space {Op} a manifold? Are there any conditions for it to be
> a manifold? If it is then we must have a fibre bundle E->M={Op}
> where the projection takes each point p on E into its orbit
> [p]=Op={q:q=gp,g in G}.
[...]
> Is there an elegant article on this topic using differential geometry?

Sounds as if you might profit from having a look at the book

R. Coquereaux & A. Jadczyk,
RIEMANNIAN GEOMETRY, FIBER BUNDLES, KALUZA-KLEIN THEORIES AND ALL THAT.....

http://www.worldscibooks.com/physics/0488.html

The table of contents lists most of what you mentioned:

-Generalities
-Riemannian Geometry of Lie Groups
-Riemannian Geometry of Homogeneous Spaces
-Riemannian Geometry of a (Right) Principal Bundle
-Riemannian Geometry of a Bundle with Fibers G/H and a Given Action of a
Lie -Group G
-Geometry of Matter Fields
-Harmonic Analysis and Dimensional Reduction
-Dimensional Reduction of the Orthogonal Bundle and of the Spin Bundles
-G-Invariance of Einstein-Yang-Mills Systems
-Action of a Bundle of Groups
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