Weak Dual Pairs in Dirac-Jacobi Geometry

TL;DR

This paper introduces the concept of weak dual pairs in Dirac-Jacobi geometry, establishing their properties, equivalence relations, and leaf correspondence theorems, with applications to Lie groupoids and alternative proofs of existing theorems.

Contribution

It defines weak dual pairs in Dirac-Jacobi geometry and proves their key properties, including an equivalence relation and leaf correspondence, extending known results in symplectic and contact geometry.

Findings

Weak dual pairs form an equivalence relation on Dirac-Jacobi manifolds.

Existence of self-dual pairs leads to an alternative proof of the normal form theorem.

Characteristic and presymplectic leaf correspondence theorems are established for weak dual pairs.

Abstract

Adopting the omni-Lie algebroid approach to Dirac-Jacobi structures, we propose and investigate a notion of weak dual pairs in Dirac-Jacobi geometry. Their main motivating examples arise from the theory of multiplicative precontact structures on Lie groupoids. Among other properties of weak dual pairs, we prove two main results. 1) We show that the property of fitting in a weak dual pair defines an equivalence relation for Dirac-Jacobi manifolds. So, in particular, we get the existence of self-dual pairs and this immediately leads to an alternative proof of the normal form theorem around Dirac-Jacobi transversals. 2) We prove the characteristic leaf correspondence theorem for weak dual pairs paralleling and extending analogous results for symplectic and contact dual pairs. Moreover, the same ideas of this proof apply to get a presymplectic leaf correspondence for weak dual pairs in Dirac geometry (not yet present in literature).