Removable presymplectic singularities and the local splitting of Dirac structures

TL;DR

This paper introduces the concept of removable presymplectic singularities, provides criteria for their identification, and proves a local splitting theorem for Dirac structures near such singularities, with applications to log-Dirac structures.

Contribution

It defines removable singularities for presymplectic forms, establishes a criterion based on regularizing functions, and proves a local splitting theorem for Dirac structures around these singularities.

Findings

Removable singularities are characterized as poles where the form's norm is unbounded.

Near removable singularities, Dirac structures split into regular and singular parts.

Log-Dirac structures generalize log-symplectic structures with new singularity behavior.

Abstract

We call a singularity of a presymplectic form $\omega$ removable in its graph if its graph extends to a smooth Dirac structure over the singularity. An example for this is the symplectic form of a magnetic monopole. A criterion for the removability of singularities is given in terms of regularizing functions for pure spinors. All removable singularities are poles in the sense that the norm of $\omega$ is not locally bounded. The points at which removable singularities occur are the non-regular points of the Dirac structure for which we prove a general splitting theorem: Locally, every Dirac structure is the gauge transform of the product of a tangent bundle and the graph of a Poisson structure. This implies that in a neighborhood of a removable singularity $\omega$ can be split into a non-singular presymplectic form and a singular presymplectic form which is the partial inverse of a Poisson bivector that vanishes at the singularity. An interesting class of examples is given by log-Dirac structures which generalize log-symplectic structures. The analogous notion of removable singularities of Poisson structures is also studied.