# nLab T-fold

Contents

### Context

#### Riemannian geometry

Riemannian geometry

duality

# Contents

## Idea

A T-fold (Hull 04) is supposed to be a kind of space that locally has charts which are Riemannian manifolds equipped with a B-field (i.e. a circle 2-bundle with connection or bundle gerbe with connection) but where the charts are glued together not just by diffeomorphisms (as an ordinary 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, due due duality in string theory. 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, as the correspondence spaces considered in topological T-duality. In the rational/infinitesimal approximation this is derived from analysis of super p-brane WZ-terms in FSS 16. A proposal for a (non-supersymmetric) global description of T-folds as total spaces of principal 2-bundles for the T-duality 2-group is in Nikolaus-Waldorf 18.

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

## References

### General

The original idea:

The relation to double field theory goes back to:

Further developments:

Discussion for nonabelian T-duality:

### Via higher geometry

A global definition of T-folds (with T-duality understood as topological T-duality) in higher differential geometry, concretely as principal 2-bundles for the T-duality 2-group (as described in T-Duality and Differential K-Theory) is proposed in

The local superspace supergeometry of T-folds compatible with this proposal is derived from fundamental super $p$-brane structure in:

The automorphism 2-group of the T-duality 2-group (and hence potentially the full structure of T-folds) is determined in:

Introduction and review:

Proposals for further generalization (2-connections and further non-geometric backgrounds):

Last revised on July 19, 2022 at 07:57:57. See the history of this page for a list of all contributions to it.