Dirac-Jacobi Bundles

TL;DR

This paper introduces Dirac-Jacobi structures on line bundles as a unifying framework that generalizes several geometric structures, and shows their integration into precontact groupoids, connecting existing results in a broader context.

Contribution

It defines Dirac-Jacobi structures via omni-Lie algebroids, unifies various geometric structures, and proves their integration into precontact groupoids.

Findings

Dirac-Jacobi structures generalize Wade's $\\mathcal{E}^1 (M)$-Dirac structures.

Integrable Dirac-Jacobi structures on line bundles integrate to precontact groupoids.

Provides a conceptual framework linking multiple existing geometric results.

Abstract

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) precontact groupoids. This puts in a conceptual framework several results already available in literature for $\mathcal E^1 (M)$-Dirac structures.