Algebroid Structures on Para-Hermitian Manifolds

Abstract

We present a global construction of a so-called D-bracket appearing in the physics literature of Double Field Theory (DFT) and show that if certain integrability criteria are satisfied, it can be seen as a sum of two Courant algebroid brackets. In particular, we show that the local picture of the extended space-time used in DFT fits naturally in the geometrical framework of para-Hermitian manifolds and that the data of an (almost) para-Hermitian manifold is sufficient to construct the D-bracket. Moreover, the twists of the bracket appearing in DFT can be interpreted in this framework geometrically as a consequence of certain deformations of the underlying para-Hermitian structure.

arXiv comment31 pages; journal version - example added, minor corrections

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Cited by in corpus (27)

• Doubled aspects of generalised dualities and integrable deformations
• Double Field Theory and Membrane Sigma-Models
• A Unique Connection for Born Geometry
• Algebroid Structures on Para-Hermitian Manifolds
• Para-Hermitian Geometries for Poisson-Lie Symmetric $\sigma$-models
• Doubled Aspects of Vaisman Algebroid and Gauge Symmetry in Double Field Theory
• Lie Algebroids and the Geometry of Off-shell BRST
• Double field theory for the A/B-models and topological S-duality in generalized geometry
• Born Sigma-Models for Para-Hermitian Manifolds and Generalized T-Duality
• Born sigma model for branes in exceptional geometry
• Global Aspects of Doubled Geometry and Pre-rackoid
• Currents, charges and algebras in exceptional generalised geometry
• Double field theory algebroid and curved $L_{\infty }$-algebras
• Global Double Field Theory is Higher Kaluza-Klein Theory
• More on Doubled Aspects of Algebroids in Double Field Theory
• Hitchhiker's Guide to Courant Algebroid Relations
• T-Dualities and Doubled Geometry of the Principal Chiral Model
• Double Field Theory and Geometric Quantisation
• 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
• The Algebroid Structure of Double Field Theory
• The puzzle of global Double Field Theory: open problems and the case for a Higher Kaluza-Klein perspective
• Extended Doubled Structures of Algebroids for Gauged Double Field Theory
• Commuting Pairs, Generalized para-Kähler Geometry and Born Geometry
• Born Geometry in a Nutshell
• Courant algebroid lifts and curved Courant algebroids
• Extended Field Theories as higher Kaluza-Klein theories