Dorfman connections and Courant algebroids

Abstract

We define Dorfman connections, which are to Courant algebroids what connections are to Lie algebroids. We illustrate this analogy with examples. In particular, we study horizontal spaces in the standard Courant algebroids over vector bundles:A linear connection ∇:X(M)×Γ(E)→Γ(E) on a vector bundle E over a smooth manifold M is tantamount to a linear splitting TE≃TqEE⊕H∇, where TqEE 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∇ to be integrable. We extend this classical result by showing that linear horizontal complements to TqEE⊕(TqEE)∘ in TE⊕T⁎E can be described in the same manner via a certain class of Dorfman connections Δ:Γ(TM⊕E⁎)×Γ(E⊕T⁎M)→Γ(E⊕T⁎M). Similarly to the tangent bundle case, we find that, after the choice of such a linear splitting, the standard Courant algebroid structure of TE⊕T⁎E→E can be completely described by properties of the Dorfman connection. As a corollary, we find that the horizontal space is a Dirac structure if and only if Δ is the dual derivation to a Lie algebroid structure on TM⊕E⁎.We use this to study splittings of TA⊕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⊕T⁎A and the linear Lie algebroid TA⊕T⁎A→TM⊕A⁎. We characterize VB- and LA-Dirac structures in TA⊕T⁎A via Dorfman connections.