Riemannian geometry

String theory




A T-fold (Hull 04) is supposed to be a kind of space that locally looks like a Riemannian manifold equipped with a B-field, but which is glued together from these not just by diffeomorphisms (as a smooth manifold is) but also by T-duality transformations along some torus fibers.

The idea is that a T-fold is a target space for a string sigma-model that is only locally a Riemannian manifold but globally a more general kind of geometry. In the literature sometimes the term non-geometric backgrounds is used for such “generalized geometric” backgrounds.

It is expected that T-folds should have a description in terms of spaces that locally are fiber products of one torus fiber bundle with its T-dual. (One proposed formalization is that these are the total spaces of principal 2-bundles for the T-duality 2-group).

One may then consider local field theory on these double torus fibrations, and this is, or is closely related to, what is called double field theory (Hull 06).

A mathematical formlization of the idea of T-folds, in terms of principal 2-bundles for the T-duality 2-group (hence in higher differential geometry) is developed in Nikolaus-Waldorf 18.


The idea was originally introduced in

The relation to double field theory goes back to

Further developments are in

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

Discussion for nonabelian T-duality:

Last revised on June 6, 2019 at 02:55:55. See the history of this page for a list of all contributions to it.