# T-folds and non-geometric backgrounds

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Recently I attended a talk where the speaker introduced and used some notions of T-folds as non-geometric backgrounds. Although I had heard about doubled geometry, double field theory and about doing physics at both $T_p\mathcal{M}$ and $T_p^*\mathcal{M}$ I knew no details. One of the astonishing things I saw was that it is possible to construct some generalized metric that includes both $G_{\mu \nu}$ and $B_{\mu \nu}$. This metric is defined on some generalization of a manifold, the T-fold. Ok, what is a T-fold?

One thing I would also like to understand is how exactly one constructs this generalized metric from the two fields above? Also, once one has this generalized metric, is it used as a "generalized background"? I understand that if the B filed is turned of one recovers classical geometry but I struggle to understand at an intuitive level what this T-geometry really is. How does this construction help us to do physics and what physics one does in these generalized backgrounds?

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Like a manifold is something obtained by using diffeomorphisms to glue Cartesian spaces together,to something that locally is still an R^n, but not globally anymore, so a T-fold is supposed to be something obtained by using diffeomorphisms and T-duality transformations to glue together type II-spacetime manifolds to something that locally is still a manifold, but not globally anymore

The rough idea and the terminology is due to

further developed in

An actual mathematical definition is still to be published, but some ideas have been developing.

A precise global definition of T-folds as principal 2-bundles for the T-duality 2-group described in the nLab entry T-Duality and Differential K-Theory is given in

• Thomas NikolausT-Duality in K-theory and elliptic cohomology, talk at String Geometry Network Meeting, Feb 2014, ESI Vienna (website)
answered Dec 9, 2014 by (5,900 points)
Hi Urs, thanks for the answer. I am aware of the nLab article but I was hopping for something more specific than what it says there at the same time without having to look at the references since I have no time :) Still thanks

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