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A coisotropic embedding theorem for pre-multisymplectic manifolds

Luca Schiavone

TL;DR

The paper proves a coisotropic embedding theorem for pre-multisymplectic manifolds, constructing a multisymplectic thickening $\tilde{\mathcal{M}}={\Lambda^{k-1}}^{\perp}_{R}(\mathcal{M})$ and embedding $\mathcal{M}$ as a $(k-1)$-coisotropic submanifold with $\mathfrak{i}^*\tilde{\omega}=\omega$. This framework yields a nondegenerate ambient multisymplectic form without requiring tubular neighborhoods or flat connections, and it clarifies the gauge-fixing interpretation via additional fields on the thickened space. An explicit 2D scalar field example demonstrates how the regularization replaces gauge freedom with extra, non-physical fields and yields explicit equations of motion. The work outlines future applications to field theories, implicit PDE integrability, and globalizing local results, offering a finite-dimensional alternative to infinite-dimensional regularization techniques.

Abstract

We prove a coisotropic embedding theorem à là Gotay for pre-multisymplectic manifolds.

A coisotropic embedding theorem for pre-multisymplectic manifolds

TL;DR

The paper proves a coisotropic embedding theorem for pre-multisymplectic manifolds, constructing a multisymplectic thickening and embedding as a -coisotropic submanifold with . This framework yields a nondegenerate ambient multisymplectic form without requiring tubular neighborhoods or flat connections, and it clarifies the gauge-fixing interpretation via additional fields on the thickened space. An explicit 2D scalar field example demonstrates how the regularization replaces gauge freedom with extra, non-physical fields and yields explicit equations of motion. The work outlines future applications to field theories, implicit PDE integrability, and globalizing local results, offering a finite-dimensional alternative to infinite-dimensional regularization techniques.

Abstract

We prove a coisotropic embedding theorem à là Gotay for pre-multisymplectic manifolds.

Paper Structure

This paper contains 7 sections, 1 theorem, 82 equations.

Table of Contents

  1. Preliminaries
  2. Coisotropic embeddings of pre-multisymplectic manifolds
  3. Application: A Scalar Field Theory with Anisotropic Mass
  4. Equations of motion
  5. Applications to Field Theories
  6. Application to the Integrability of Implicit Partial Differential Equations
  7. Extending Local Results to Global Contexts

Key Result

Theorem 2.1

Let $(\mathcal{M}, \omega)$ be a pre-$k$-plectic manifold, with $k > 2$. Assume $K \,=\, \mathrm{ker}\, \omega$ to have constant rank. Then, there exists a multisymplectic manifold $\tilde{\mathcal{M}}$, referred to as the multisymplectic thickening of $\mathcal{M}$, and an embedding $\mathfrak{i} \

Theorems & Definitions (11)

  • Definition 1.1: Multisymplectic manifold
  • Definition 1.2: Pre-multisymplectic manifold
  • Definition 1.3: $\ell$-multisymplectic orthogonal Cantrijn-Ibort-DeLeon-Premultisymplectic-1999
  • Definition 1.4: $\ell$-Coisotropic submanifold Cantrijn-Ibort-DeLeon-Premultisymplectic-1999
  • Definition 1.5: Bundle of $k$-forms
  • Definition 1.6: Connection on $\mathcal{M}$
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • ...and 1 more