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The notes for my class on Kaluza-Klein reduction are a bit all over the place and at times it's difficult to follow what's going on. (I plan on asking a specific question about an example later).

For now, are there any resources that introduce Kaluza-Klein reduction with some worked examples, apart from the simple Minkowski space with a circle attached. I'd like to see some explicit examples where we have a higher dimensional torus for example.

A treatment including supergravity would be a plus.

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    $\begingroup$ How about any ol' string book: Compactifications are everywhere! $\endgroup$ – Danu Jan 28 '16 at 11:18
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Did you had a look at this one KALUZA-KLEIN SUPERGRAVITY ? (and references therein)

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Chis Pope's notes are very useful - http://people.physics.tamu.edu/pope/ihplec.pdf

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