Graded Poisson and Graded Dirac structures

TL;DR

This paper reviews and unifies graded symplectic, Poisson, and Dirac structures, proposing a comprehensive generalization to enhance the geometric framework for classical field theories and their quantization.

Contribution

It introduces a new graded generalization that encompasses existing structures, facilitating progress in classical field theories and related areas.

Findings

Unified framework for graded structures

Potential applications in classical field theories

Foundation for future quantization methods

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.