abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
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 is glued together from these not just by diffeomorphisms 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).
The idea was originally introduced in
The relation to double field theory goes back to
Further developments are in
Chris Hull, Global Aspects of T-Duality, Gauged Sigma Models and T-Folds (arXiv:hep-th/0604178)
Aaron Bergman, Daniel Robbins, Ramond-Ramond Fields, Cohomology and Non-Geometric Fluxes (arXiv:0710.5158)
Dieter Lüst, Stefano Massai, Valentí Vall Camell, The monodromy of T-folds and T-fects (arXiv:1508.01193)
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