Dirac Surfaces and Threefolds

TL;DR

This paper classifies Dirac structures on surfaces and 3-manifolds, showing they are described by foliations, circle bundles, presymplectic, and Poisson structures, providing a comprehensive geometric characterization.

Contribution

It offers a complete description of Dirac structures on low-dimensional manifolds, linking them to well-understood geometric entities and expanding the understanding of their structure.

Findings

Dirac structures on surfaces correspond to regular 1-foliations or circle bundle sections.

On 3-manifolds, Dirac structures are unions of presymplectic and foliated Poisson or vice versa.

Provides a unified geometric framework for understanding Dirac structures in low dimensions.

Abstract

We describe Dirac structures on surfaces and 3-manifolds. Every Dirac structure on a surface $M$ is described either by a regular 1-foliation or by a section of a circle bundle obtained as a fiberwise compactification of the line bundle $\wedge^2TM$. Every Dirac structure on a 3-manifold $M$ is either the union of a presymplectic manifold and a foliated Poisson manifold, or the union of a Poisson manifold and a foliated presymplectic manifold.