Natural lifts of Dorfman brackets

Let $E$ a smooth vector bundle over a smooth manifold $M$. This note proves that a Dorfman bracket on $TM \oplus E^{\ast}$, anchored by $\mathrm{pr}_{TM}$, is equivalent to a lift from $\Gamma (TM \oplus E^{\ast})$ to linear sections of $TE \oplus T^{\ast} E \to E$, that intertwines the given Dorfman bracket with the Courant–Dorfman bracket on sections of $TE \oplus T^{\ast} E$. This shows a universality of the Courant–Dorfman bracket, and allows us to characterise twistings and symmetries of transitive Dorfman brackets via the corresponding lifts.

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During the completion of this work, MJL was supported by a Vice-Chancellor’s fellowship from The University of Sheffield. CKL was supported by an STFC Studentship and a Graduate Studentship from Trinity College, Cambridge.

Published 3 May 2019