Geometric analysis for the Pontryagin action and boundary terms

TL;DR

This work addresses boundary terms and differentiability for the Pontryagin action by deploying covariant geometric formalisms. It develops and compares Lagrangian, multisymplectic, and polysymplectic formulations and uses space-time decomposition to connect covariant data with instantaneous dynamics, gauge generators, and boundary conditions. A key finding is that differentiability conditions on manifolds with boundary emerge naturally from Noether theorems across all formalisms, with the De Donder–Weyl Hamiltonian and Poisson–Gerstenhaber structures providing a consistent covariant Hamiltonian picture. The analysis confirms that the Pontryagin model is topological, possessing only first-class constraints and a vanishing t-instantaneous Hamiltonian on the constraint surface, and clarifies how boundary terms relate to edge dynamics and boundary invariants such as Chern–Simons terms.

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.