Abstract
Abstract We give a self-contained survey of some approaches aimed at a global description of the geometry underlying double field theory. After reviewing the geometry of Courant algebroids and their incarnations in the AKSZ construction, we develop the theory of metric algebroids including their graded geometry. We use metric algebroids to give a global description of doubled geometry, incorporating the section constraint, as well as an AKSZ-type construction of topological doubled sigma-models. When these notions are combined with ingredients of para-Hermitian geometry, we demonstrate how they reproduce kinematical features of double field theory from a global perspective, including solutions of the section constraint for Riemannian foliated doubled manifolds, as well as a natural notion of generalized T-duality for polarized doubled manifolds. We describe the L ∞-algebras of symmetries of a doubled geometry, and briefly discuss other proposals for global doubled geometry in the literature.
Highlights
This contribution is a relatively self-contained survey of some mathematical approaches to a rigorous global formulation of the geometry underlying double eld theory, that we will colloquially call ‘doubled geometry’, following standard terminology from string theory
When these notions are combined with ingredients of para-Hermitian geometry, we demonstrate how they reproduce kinematical features of double eld theory from a global perspective, including solutions of the section constraint for Riemannian foliated doubled manifolds, as well as a natural notion of generalized T-duality for polarized doubled manifolds
We introduce general notions of algebroids, culminating in Lie algebroids and Courant algebroids. We develop their formulations as symplectic Lie n-algebroids in graded geometry and the corresponding AKSZ sigma-models, together with their gauge symmetries which can be formulated in terms of at L∞-algebras
Summary
Double eld theory is an extension of supergravity in which stringy T-duality becomes a manifest symmetry. The basic example of a doubled geometry in this context comes from considering toroidal compacti cations of string theory, which we shall brie y review
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