T-Duality, Non-Geometricity & Torus Fibrations

[fra87]

Hi!

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!

 

[AndyW]

Hello!

It's great to see that you are studying T-duality and non-geometricity in string theory. These are very interesting and important topics in modern theoretical physics. In response to your question, you are correct that the free and transitive action of globally defined Killing vectors defines a principal torus bundle T^n. This is known as a torus fibration and it is a key concept in T-duality.

When the Killing vectors are only locally defined, it leads to an affine torus bundle. This means that the base space of the bundle is not just a torus, but a more general space called an affine torus. The affine torus is a higher dimensional space that contains the usual torus as a submanifold.

The reason a non-trivial monodromy comes out in this case is because the affine torus bundle has non-trivial topology. This means that when you go around a closed loop in the base space, the fiber of the bundle will not return to its original position. This results in a monodromy, which is a transformation that relates the different fibers of the bundle.

An explicit example of this can be seen in the study of T-folds, which are non-geometric backgrounds in string theory. These backgrounds are described by an affine torus bundle with non-trivial monodromy. I would recommend looking into the work of O. Hull and B. Zwiebach on T-folds for more information.

Another helpful reference for understanding the relation between T-duality and gauging of a sigma model is the paper "T-duality and non-geometric backgrounds in string theory" by A. Dabholkar and C. Hull.

I hope this helps to clarify your understanding. Good luck with your studies!