Courant algebroid lifts and curved Courant algebroids
Filip Moučka, Roberto Rubio
TL;DR
This work develops the Courant algebroid lift, producing curved Courant algebroids on the total space of a vector bundle from a base Courant-like structure and a connection, and introduces curved Cartan calculus to organize curvature via $T\in\Omega^2(M,TM)$. It proves a precise hierarchy of CA-like structures, establishes a complete classification of exact curved CA in terms of a pair $(T,H)$ with $H\in\Omega^3(M)$ modulo $\mathrm{im}\,\mathrm{d}^{T}$, and shows that the lift preserves or augments CA properties depending on the curvature of the underlying connection. The paper then analyzes numerous examples: the fundamental Courant algebroid leading to Patterson–Walker metrics, lifts related to Poisson structures, $B_n$-geometry, and quadratic Lie algebras, and demonstrates how lifts yield wide classes of Courant algebroid actions with coisotropic stabilizers. Overall, the Courant algebroid lift provides a unifying framework connecting generalized geometry, curved DG Lie algebras, and diverse geometric structures via explicit lifts and curvature constraints, with broad potential applications in geometry and mathematical physics.
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.