Courant algebroid lifts and curved Courant algebroids

TL;DR

This paper introduces the concept of Courant algebroid lifts, extending the structure to curved Courant algebroids, and explores their classification, connections to geometry, and applications in actions and generalized geometry.

Contribution

It defines curved Courant algebroids, establishes their properties, classifies exact cases, and links lifts to various geometric structures and Lie algebra actions.

Findings

Classification of exact curved Courant algebroids.

Link between Courant algebroid lifts and generalized geometry.

Relation of lifts to Lie algebras, Poisson, and complex geometry.

Abstract

We introduce the Courant algebroid lift, a new construction that takes a Courant algebroid together with a vector bundle connection and produces, when the connection is flat in the image of the anchor, a Courant algebroid. In general, this lift produces a Courant-like structure that we call a curved Courant algebroid. We start by establishing a hierarchy of Courant algebroid properties and their associated structures. In this setting, we introduce curved Courant algebroids, which we show to be related to connections with torsion and curved differential graded Lie algebras. We use this to provide a classification of exact curved Courant algebroids. We show that the Courant algebroid lift of an exact Courant algebroid yields a natural link between the Patterson-Walker metric and generalized geometry. By lifting non-exact Courant algebroids, we establish a relation of these lifts to Lie algebras, Poisson and special complex geometry. Finally, we show that Courant algebroid lifts provide a large class of examples of Courant algebroid actions.