Extended Riemannian Geometry III: Global Double Field Theory with Nilmanifolds

Abstract

We describe the global geometry, symmetries and tensors for Double Field Theory over pairs of nilmanifolds with fluxes or gerbes. This is achieved by a rather straightforward application of a formalism we developed previously. This formalism constructs the analogue of a Courant algebroid over the correspondence space of a T-duality, using the language of graded manifolds, derived brackets and we use the description of nilmanifolds in terms of periodicity conditions rather than local patches. The strong section condition arises purely algebraically, and we show that for a particularly symmetric solution of this condition, we recover the Courant algebroids of both nilmanifolds with fluxes. We also discuss the finite, global symmetries of general local Double Field Theory and explain how this specializes to the case of T-duality between nilmanifolds.

arXiv comment36 pages, typos fixed, references added, presentation improved and discussion added, published version

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Cited by in corpus (6)

• Remarks on the non-Riemannian sector in Double Field Theory
• Global Double Field Theory is Higher Kaluza-Klein Theory
• A note on Faddeev--Popov action for doubled-yet-gauged particle and graded Poisson geometry
• Towards an extended/higher correspondence -- Generalised geometry, bundle gerbes and global Double Field Theory
• Higher curvature Bianchi identities, generalised geometry and $L_{\infty }$ algebras
• The puzzle of global Double Field Theory: open problems and the case for a Higher Kaluza-Klein perspective