How Are Maurer-Cartan Forms Utilized in Physics?

[haushofer]

Hi,

I'm trying to understand the use of Maurer-Cartan one-forms in physics. As far as I understand it's a Lie-algebra valued one-form which sends vectors at an arbitrary point g on the Lie-group to the identity e (the Lie algebra). But my question is: what is the use of these things in physics? I have the feeling that somehow they let you construct metrics for spaces with symmetries described by the Lie-group in question, but can someone elaborate on this or give some references where people explain this?

 

[arkajad]

More fundamental than the invariant metric is the parallelism of the group manifold.

 

[haushofer]

Could you be a bit more specific? :)

 

[arkajad]

Call the Maurer-Cartan form $\omega$. Take a given vector $\xi$ in the Lie algebra of the group. At each point g of the group there is a unique tangent vector $\xi_p$ with the property that $\omega(\xi_p)=\xi.$. This way you create vector fields on the group - one vector field for each element of the Lie algebra. This defines global parallelism. You can compare tangent vectors at a distance by requiring that $\omega$ has the same value on both vectors. It defines globally flat affine connection on the group manifold - it has vanishing curvature, but non-vanishing torsion.

 

[haushofer]

Ok, that's clear. And how does this manifest itself in physics? "Given a Lie algebra, one can construct the corresponding space with that isometry" or something?

 

[arkajad]

In physics we are usually dealing with homogeneous spaces G/H. Their geometry is more complicated than that of the group itself, though geometry of G plays a role there too. The first nice example to look at is the two-sphere, the homogeneous space SO(3)/SO(2). It has an invariant metric, but it is not parallelizable.
For a use of Maurer-Cartan forms within the framework of homogeneous space you may like to check http://books.google.fr/books?id=zGp...&q="cartan-maurer" homogeneous space&f=false".

 

[haushofer]

Thanks for that link! I will definitely check it, and if I have more questions I'll come back! :)