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Graded Poisson and Graded Dirac structures

Manuel de León, Rubén Izquierdo-López

TL;DR

The paper addresses the problem of unifying generalized Dirac/Poisson geometries for classical field theories by introducing a graded framework that extends multisymplectic and higher Dirac notions. It develops both linear and manifold-level graded Dirac structures, proves an equivalence between linear graded Dirac structures and linear weak higher Dirac structures, and defines graded Poisson structures with a compatible graded bracket on Hamiltonian forms, including a multisymplectic foliation and a bracket–structure correspondence. The key contributions include a precise graph description of graded Dirac data, an equivalence result linking linear graded Dirac and weak higher Dirac structures, and the construction of a graded Poisson bracket that recovers dynamics and currents while inducing foliations by multisymplectic leaves. The framework offers a robust foundation for analyzing integrability, reduction, currents, and potential quantization in classical field theories, under mild regularity assumptions, thereby enabling broader applications to higher-form symmetries and field-theoretic quantization.

Abstract

There have been several attempts in recent years to extend the notions of symplectic and Poisson structures in order to create a suitable geometrical framework for classical field theories, trying to achieve a success similar to the use of these concepts in Hamiltonian mechanics. These notions always have a graded character, since the multisymplectic forms are of a higher degree than two. Another line of work has been to extend the concept of Dirac structures to these new scenarios. In the present paper we review all these notions, relate them and propose and study a generalization that (under some mild regularity conditions) includes them and is of graded nature. We expect this generalization to allow us to advance in the study of classical field theories, their integrability, reduction, numerical approximations and even their quantization.

Graded Poisson and Graded Dirac structures

TL;DR

The paper addresses the problem of unifying generalized Dirac/Poisson geometries for classical field theories by introducing a graded framework that extends multisymplectic and higher Dirac notions. It develops both linear and manifold-level graded Dirac structures, proves an equivalence between linear graded Dirac structures and linear weak higher Dirac structures, and defines graded Poisson structures with a compatible graded bracket on Hamiltonian forms, including a multisymplectic foliation and a bracket–structure correspondence. The key contributions include a precise graph description of graded Dirac data, an equivalence result linking linear graded Dirac and weak higher Dirac structures, and the construction of a graded Poisson bracket that recovers dynamics and currents while inducing foliations by multisymplectic leaves. The framework offers a robust foundation for analyzing integrability, reduction, currents, and potential quantization in classical field theories, under mild regularity assumptions, thereby enabling broader applications to higher-form symmetries and field-theoretic quantization.

Abstract

There have been several attempts in recent years to extend the notions of symplectic and Poisson structures in order to create a suitable geometrical framework for classical field theories, trying to achieve a success similar to the use of these concepts in Hamiltonian mechanics. These notions always have a graded character, since the multisymplectic forms are of a higher degree than two. Another line of work has been to extend the concept of Dirac structures to these new scenarios. In the present paper we review all these notions, relate them and propose and study a generalization that (under some mild regularity conditions) includes them and is of graded nature. We expect this generalization to allow us to advance in the study of classical field theories, their integrability, reduction, numerical approximations and even their quantization.
Paper Structure (16 sections, 37 theorems, 306 equations)

This paper contains 16 sections, 37 theorems, 306 equations.

Table of Contents

  1. Introduction
  2. An overview of previous generalizations of Dirac structures to classical field theories
  3. Multisymplectic manifolds
  4. Dirac structures
  5. Multi-Dirac structures
  6. Higher Dirac structures
  7. Weak higher Dirac structures
  8. Graded Dirac structures
  9. Linear graded Dirac structures
  10. Graded Dirac manifolds
  11. Graded Poisson structures
  12. Graded Poisson manifolds
  13. Graded Poisson brackets
  14. Currents and conserved quantities
  15. Conclusions and further work
  16. ...and 1 more sections

Key Result

Proposition 2.3.1

Let $\omega \in \Omega^{k+1}(M)$ be a $(k+1)$-form. Then, the sequence of vector subbundles defines a multi-Dirac structure of order $k$ if and only if $\operatorname{d}\!{\omega} = 0.$

Theorems & Definitions (109)

  • Definition 1.0.1: Higher Poisson structure
  • Definition 2.1.1: Multisymplectic manifold
  • Example 2.1.1
  • Example 2.1.2
  • Definition 2.1.2: Hamiltonian multivector field, Hamiltonian form
  • Definition 2.2.1: Dirac structure
  • Definition 2.3.1: Multi-Dirac structure
  • Proposition 2.3.1: Vankerschaver2011
  • proof
  • Definition 2.4.1: Higher Dirac structure
  • ...and 99 more