Geometric analysis for the Pontryagin action and boundary terms

TL;DR

This paper provides a comprehensive geometric analysis of the Pontryagin action, focusing on boundary conditions, symmetries, and covariant momentum maps, and demonstrates how boundary terms naturally emerge from Noether's theorem in various formalisms.

Contribution

It introduces a unified geometric framework for analyzing the Pontryagin model, emphasizing boundary conditions and the role of symmetries across multiple covariant and Hamiltonian formalisms.

Findings

Boundary conditions are derived from Noether's theorem in geometric formulations.

The covariant momentum map extends Noether's theorems to boundary terms.

The analysis recovers the Hamiltonian and gauge generators via space-time decomposition.

Abstract

In this article, we analyze the Pontryagin model adopting different geometric-covariant approaches. In particular, we focus on the manner in which boundary conditions must be imposed on the background manifold in order to reproduce an unambiguous theory on the boundary. At a Lagrangian level, we describe the symmetries of the theory and construct the Lagrangian covariant momentum map which allows for an extension of Noether's theorems. Through the multisymplectic analysis we obtain the covariant momentum map associated with the action of the gauge group on the covariant multimomenta phase-space. By performing a space plus time decomposition by means of a foliation of the appropriate bundles, we are able to recover not only the $t$-instantaneous Lagrangian and Hamiltonian of the theory, but also the generator of the gauge transformations. In the polysymplectic framework we perform a Poisson-Hamilton analysis with the help of the De Donder-Weyl Hamiltonian and the Poisson-Gerstenhaber bracket. Remarkably, as long as we consider a background manifold with boundary, in all of these geometric formulations, we are able to recover the so-called differentiability conditions as a straightforward consequence of Noether's theorem.