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I'm studying questions concerning T-duality and non-geometricity in string theory. In particular i'm focusing on the relation between T-duality and gauging of a sigma model.

It's known that the free and transitive action of a set of globally defined (and commuting) Killing vectors defines a principal torus bundle T^n. In this case the holonomy should be encoded by the path ordered integral of the exponential of a local connection. Is it true?

I actually don't understand what happens when the Killing vectors are only locally defined. Why this gives rise to an affine torus bundle, and why a non-trivial monodromy comes out?

Can you explain me with an explicit example, or can you indicate me some clear references?

Thanks!