Dorfman connections and Courant algebroids

TL;DR

This paper introduces Dorfman connections as a new framework to understand Courant algebroids, providing a unified approach to describe their structure, splittings, and related geometric objects.

Contribution

It defines Dorfman connections for Courant algebroids, linking them to linear splittings and describing the Courant structure through these connections, with applications to Lie algebroids and Dirac structures.

Findings

Dorfman connections characterize Courant algebroid splittings.

The Courant algebroid structure can be described via properties of Dorfman connections.

Applications include characterizations of VB- and LA-Dirac structures.

Abstract

We define Dorfman connections, which are to Courant algebroids what connections are to Lie algebroids. Several examples illustrate this analogy. A linear connection $\nabla\colon \mathfrak{X}(M)\times\Gamma(E)\to\Gamma(E)$ on a vector bundle $E$ over a smooth manifold $M$ is tantamount to a linear splitting $TE\simeq T^{q_E}E\oplus H_\nabla$, where $T^{q_E}E$ is the set of vectors tangent to the fibres of $E$. Furthermore, the curvature of the connection measures the failure of the horizontal space $H_\nabla$ to be integrable. We show that linear horizontal complements to $T^{q_E}E\oplus (T^{q_E}E)^\circ$ in the Pontryagin bundle over the vector bundle $E$ can be described in the same manner via a certain class of Dorfman connections $\Delta\colon \Gamma(TM\oplus E^*)\times\Gamma(E\oplus T^*M)\to\Gamma(E\oplus T^*M)$. Similarly to the tangent bundle case, we find that, after the choice of a linear splitting, the standard Courant algebroid structure of $TE\oplus T^*E\to E$ can be completely described by properties of the Dorfman connection. As an application, we study splittings of $TA\oplus T^*A$ over a Lie algebroid $A$ and, following Gracia-Saz and Mehta, we compute the representations up to homotopy defined by any linear splitting of $TA\oplus T^*A$ and the linear Lie algebroid $TA\oplus T^*A\to TM\oplus A^*$. Further, we characterise VB- and LA-Dirac structures in $TA\oplus T^*A$ via Dorfman connections.