nLab T-fold

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).

Related concepts

• type II geometry

• non-geometric string theory vacua

• doubled geometry, type II geometry

• double field theory

References

General

The original idea:

• Chris Hull, A Geometry for Non-Geometric String Backgrounds, JHEP0510:065,2005 (arXiv:hep-th/0406102)

The relation to double field theory goes back to:

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

Further developments:

• 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)

• Yoan Gautier, Chris Hull, Dan Israël, Heterotic/type II Duality and Non-Geometric Compactifications (arXiv:1906.02165)

Discussion for nonabelian T-duality:

• Mark Bugden, Non-abelian T-folds (arXiv:1901.03782)

• Ladislav Hlavatý, Ivo Petr, T-folds as Poisson-Lie plurals (arXiv:2004.08387)

Comments in relation to topological T-duality:

• Peter Bouwknegt, Ashwin S. Pande, Topological T-duality and T-folds, Advances in Theoretical and Mathematical Physics 13 5 (2009) [arXiv:0810.4374, doi:10.4310/ATMP.2009.v13.n5.a6]

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

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

• Thomas Nikolaus, Konrad Waldorf, Higher geometry for non-geometric T-duals, Commun. Math. Phys. 374 (2020) 317-366 [arXiv:1804.00677, doi:10.1007/s00220-019-03496-3]

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

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, T-Duality from super Lie n-algebra cocycles for super p-branes, ATMP Volume 22 (2018) Number 5 (arXiv:1611.06536, doi:10.4310/ATMP.2018.v22.n5.a3)

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

• Konrad Waldorf, Categorical symmetries of T-duality [arXiv:2202.09187, spire:2035224]

Introduction and review:

• Konrad Waldorf, Stacks, Gerbes and T-Duality, talk notes, Adelaide (2019) [pdf]

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

• Hyungrok Kim, Christian Saemann, Non-Geometric T-Duality as Higher Groupoid Bundles with Connections [arXiv:2204.01783]