Algebroids, AKSZ Constructions and Doubled Geometry

TL;DR

This paper surveys the global geometric structures underlying double field theory, focusing on algebroids, AKSZ constructions, and doubled geometry, and introduces new perspectives on symmetries and T-duality.

Contribution

It develops the theory of metric algebroids and their graded geometry to provide a global description of doubled geometry, including solutions to the section constraint and generalized T-duality.

Findings

Global description of doubled geometry using metric algebroids

Construction of topological doubled sigma-models via AKSZ

Identification of symmetry structures with $L_$-algebras

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 $L_\infty$-algebras of symmetries of a doubled geometry, and briefly discuss other proposals for global doubled geometry in the literature.