Journal of Symplectic Geometry

Volume 16 (2018)

Number 2

Dirac–Jacobi bundles

Pages: 485 – 561



Luca Vitagliano (Dipartimento di Matematica, Università degli Studi di Salerno, Fisciano (SA), Italy; and Istituto Nazionale di Fisica Nucleare, GC Salerno, Fisciano (SA), Italy)


We show that a suitable notion of Dirac–Jacobi structure on a generic line bundle $L$, is provided by Dirac structures in the omni-Lie algebroid of $L$. Dirac–Jacobi structures on line bundles generalize Wade’s $\mathcal{E}^1(M)$-Dirac structures and unify generic (i.e. non-necessarily coorientable) precontact distributions, Dirac structures and local Lie algebras with one-dimensional fibers in the sense of Kirillov (in particular, Jacobi structures in the sense of Lichnerowicz). We study the main properties of Dirac–Jacobi structures and prove that integrable Dirac–Jacobi structures on line-bundles integrate to (non-necessarily coorientable) pre-contact groupoids. This puts in a conceptual framework several results already available in literature for $\mathcal{E}^1(M)$-Dirac structures.

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Received 26 March 2015

Accepted 27 January 2016