gmoutso

2005-04-05 17:14:23 UTC

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

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