nLab pre-symplectic current

Idea

A local Lagrangian density $L \in \Omega^{p+1,0}(E)$ on a jet bundle $J^\infty(E)$ of some field bundle $E \to \Sigma$ induces a differential form

$$\theta \in \Omega^{p,1}(E)$$

unique up to terms in the image of the horizontal derivative, given by

$$d L = \delta_{EL} L - d_H \theta \,,$$

where $\delta_{EL}$ is the Euler-Lagrange operator. Its image under the vertical derivative

$$\omega \coloneqq d_V \theta \in \Omega^{p,2}(E)$$

is called the pre-symplectic current. Accordingly $\theta$ itself is also called the presymplectic potential current.

This is because for any choice of $p$-dimensional submanifold $\Sigma_p \hookrightarrow \Sigma$, the transgression of variational differential forms of $\omega$ to $\Sigma_p$ is a presymplectic form on the on-shell space of sections on the infinitesimal neighbourhood of $\Sigma_p$. This is the covariant phase space.

References

• Higher Prequantum Geometry IV: The Covariant Phase Space – Transgressively

• Grigorios Giotopoulos, Hisham Sati, §7 in: Field Theory via Higher Geometry I: Smooth Sets of Fields [arXiv:2312.16301]