On dual pairs in Dirac Geometry

TL;DR

This paper explores dual pairs in Dirac geometry, providing a new construction of strong self-dual pairs, and offers simplified proofs and new insights into related theorems like symplectic realizations and normal forms.

Contribution

It introduces an explicit construction of strong self-dual pairs in Dirac structures, connecting them to symplectic realizations and normal form theorems with a more natural approach.

Findings

Constructed strong self-dual pairs for Dirac structures.

Provided a Dirac-theoretic version of Libermann's theorem.

Simplified proofs of symplectic realizations and normal form theorems.

Abstract

In this note we discuss dual pairs in Dirac geometry. We show that this notion appears naturally when studying the problem of pushing forward a Dirac structure along a surjective submersion, and we prove a Dirac-theoretic version of Libermann's theorem from Poisson geometry. Our main result is an explicit construction of strong self-dual pairs for Dirac structures. This theorem not only recovers the global construction of symplectic realizations from [Crainic-Marcut 2011], but allows for a more conceptual understanding of it, yielding a simpler and more natural proof. As an application of the main theorem, we present a different approach to the recent normal form theorem around Dirac transversals from [Bursztyn-Lima-Meinrenken 2016].