Advances in Theoretical and Mathematical Physics

Volume 22 (2018)

Number 6

Natural lifts of Dorfman brackets

Pages: 1401 – 1446



Madeleine Jotz Lean (Mathematisches Institut, Georg-August Universität Göttingen, Germany)

Charlotte Kirchhoff-Lukat (Departement Wiskunde, KU Leuven, Belgium)


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.

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