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Homotopy reduction of multisymplectic structures in Lagrangian field theory

Janina Bernardy

TL;DR

The paper develops a homotopy-theoretic reduction scheme for premultisymplectic structures in Lagrangian field theory by leveraging the $L_ infty$-algebra of Hamiltonian forms and the notion of homotopy momentum maps within the variational bicomplex framework. It recasts symmetry obstructions via an obstruction complex on the double/total complexes and introduces the homotopy zero locus as a field-theoretic analogue of a level set for momentum maps, ensuring invariance under symmetry actions. The method is illustrated through classical mechanics examples (space translations, rotations, and time translations) and extended to field-theoretic contexts, highlighting how conserved currents and charges arise as components of homotopy momentum maps and how reduction operates on the space of fields. The work positions homotopy reduction as a geometric alternative to existing multisymplectic reduction approaches, with potential applications to diffeomorphism symmetry in general relativity and connections to other reductions of the $L_ linfty$-algebra of Hamiltonian forms.

Abstract

While symplectic geometry is the geometric framework of classical mechanics, the geometry of classical field theories is governed by multisymplectic structures. In multisymplectic geometry, the Poisson algebra of Hamiltonian functions is replaced by the $L_\infty$-algebra of Hamiltonian forms introduced by Rogers in 2012. The corresponding notion of homotopy momentum maps as morphisms of $L_\infty$-algebras is due to Callies, Frégier, Rogers, and Zambon in 2016. We develop a method of homotopy reduction for local homotopy momentum maps in Lagrangian field theory using these homotopy algebraic structures.

Homotopy reduction of multisymplectic structures in Lagrangian field theory

TL;DR

The paper develops a homotopy-theoretic reduction scheme for premultisymplectic structures in Lagrangian field theory by leveraging the -algebra of Hamiltonian forms and the notion of homotopy momentum maps within the variational bicomplex framework. It recasts symmetry obstructions via an obstruction complex on the double/total complexes and introduces the homotopy zero locus as a field-theoretic analogue of a level set for momentum maps, ensuring invariance under symmetry actions. The method is illustrated through classical mechanics examples (space translations, rotations, and time translations) and extended to field-theoretic contexts, highlighting how conserved currents and charges arise as components of homotopy momentum maps and how reduction operates on the space of fields. The work positions homotopy reduction as a geometric alternative to existing multisymplectic reduction approaches, with potential applications to diffeomorphism symmetry in general relativity and connections to other reductions of the -algebra of Hamiltonian forms.

Abstract

While symplectic geometry is the geometric framework of classical mechanics, the geometry of classical field theories is governed by multisymplectic structures. In multisymplectic geometry, the Poisson algebra of Hamiltonian functions is replaced by the -algebra of Hamiltonian forms introduced by Rogers in 2012. The corresponding notion of homotopy momentum maps as morphisms of -algebras is due to Callies, Frégier, Rogers, and Zambon in 2016. We develop a method of homotopy reduction for local homotopy momentum maps in Lagrangian field theory using these homotopy algebraic structures.
Paper Structure (19 sections, 5 theorems, 56 equations)

This paper contains 19 sections, 5 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Content and main results
  4. Conventions
  5. Obstruction theory for Hamiltonian actions
  6. The classical case: presymplectic manifolds
  7. Premultisymplectic manifolds
  8. Space translation:
  9. Space rotation:
  10. Time translation:
  11. Reduction
  12. The classical case: (pre-)symplectic reduction at value $0$
  13. Step 1: Zero locus
  14. Step 2: Invariance of the zero locus
  15. Homotopy reduction of premultisymplectic structures in Lagrangian field theory
  16. ...and 4 more sections

Key Result

Theorem 2.2

Let $\rho: \mathfrak{g} \rightarrow \mathfrak{X}(X)$ be an infinitesimal action on the presymplectic manifold $(X, \omega)$. A primitive $\overline{\mu} \in \Omega^{1,0}(\mathfrak{g}, X)$ of $\overline{\omega}\in \Omega^{2}(\mathfrak{g}, X)$ provides a momentum map.

Theorems & Definitions (17)

  • Example 2.1: Angular momentum in classical mechanics
  • Theorem 2.2
  • Definition 2.3: Thm. 5.2 in Rogers2012 and Thm. 6.7 in Zambon2012
  • Definition 2.4: Def./Prop. 5.1 in CFRZ2016
  • Example 2.5: Symmetries in classical mechanics
  • Example 2.6: Gauge symmetry of Chern-Simons theory
  • Example 2.7: Diffeomorphism symmetry of general relativity
  • Theorem 2.8
  • proof
  • Remark 3.1
  • ...and 7 more