The Space of Integrable Dirac Structures on Hilbert C*-Modules

TL;DR

This paper studies the space of integrable Dirac structures on Hilbert C*-modules, linking integrability to automorphism groups and PDE solutions, and explores their topological properties.

Contribution

It introduces a new interpretation of integrability of Dirac structures via automorphism groups on Hilbert C*-modules, connecting geometric and algebraic perspectives.

Findings

Integrability characterized by automorphism group properties

Topological properties of the space of Dirac structures analyzed

In special cases, integrability linked to PDE solutions

Abstract

In this paper we interpret the integrability of the Dirac structures on some Hilbert C*-modules in terms of an automorphism group. This is the group of orthogonal transformations on the Hilbert C*-module of sections of a Hermitian vector bundle over an smooth manifold M. Some topological properties of the group of integrable Dirac structures are studied. In some special cases it is shown that the integrability condition corresponds to the solutions of a partial differential equation. This is explained as a necessary and sufficient condition.