Presymplectic structures and intrinsic Lagrangians

Abstract

It is well-known that a Lagrangian induces a compatible presymplectic form on the equation manifold (stationary surface, understood as a submanifold of the respective jet-space). Given an equation manifold and a compatible presymplectic form therein, we define the first-order Lagrangian system which is formulated in terms of the intrinsic geometry of the equation manifold. It has a structure of a presymplectic AKSZ sigma model for which the equation manifold, equipped with the presymplectic form and the horizontal differential, serves as the target space. For a wide class of systems (but not all) we show that if the presymplectic structure originates from a given Lagrangian, the proposed first-order Lagrangian is equivalent to the initial one and hence the Lagrangian per se can be entirely encoded in terms of the intrinsic geometry of its stationary surface. If the compatible presymplectic structure is generic, the proposed Lagrangian is only a partial one in the sense that its stationary surface contains the initial equation manifold but does not necessarily coincide with it.

arXiv comment26 pages

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Cited by in corpus (10)

• Continuous spin gauge field in (A)dS space
• Gauge PDE and AKSZ-type Sigma Models
• Presymplectic structures and intrinsic Lagrangians for massive fields
• Covariant canonical formulations of classical field theories
• Higher Spin Gravities and Presymplectic AKSZ Models
• Variational Tricomplex, Global Symmetries and Conservation Laws of Gauge Systems
• General Relativity and the AKSZ construction
• Presymplectic gauge PDEs and Lagrangian BV formalism beyond jet-bundles
• Symmetric vs. chiral approaches to massive fields with spin
• Homotopy Manin Theories: Generalising Third-Way, Yang-Mills and Integrable Sigma Models