T-folds and non-geometric backgrounds

Question

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? 

Answer 1

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 

• Chris Hull, A Geometry for Non-Geometric String Backgrounds (arXiv:hep-th/0406102)

further developed in

• Chris Hull, Doubled geometry and T-folds JHEP0707:080,2007 (arXiv:hep-th/0605149)

• Chris Hull, Global Aspects of T-Duality, Gauged Sigma Models and T-Folds (arXiv:hep-th/0604178)

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 Nikolaus, T-Duality in K-theory and elliptic cohomology, talk at String Geometry Network Meeting, Feb 2014, ESI Vienna (website)

Comments

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