Covariant Hamiltonian Field Theories on Manifolds with Boundary: Yang-Mills Theories

TL;DR

This paper extends multisymplectic formalism to covariant Hamiltonian field theories on manifolds with boundaries, providing a geometric framework for boundary conditions, phase space reduction, and quantization, with applications to Yang-Mills and other theories.

Contribution

It develops a geometric multisymplectic framework for boundary conditions in covariant Hamiltonian field theories, including gauge theories like Yang-Mills.

Findings

Boundary conditions arise from the moment map of the gauge group.

The reduced phase space is a symplectic manifold with an isotropic boundary submanifold.

The framework is validated on scalar fields, Poisson sigma-model, and Yang-Mills theories.

Abstract

The multisymplectic formalism of field theories developed by many mathematicians over the last fifty years is extended in this work to deal with manifolds that have boundaries. In particular, we develop a multisymplectic framework for first order covariant Hamiltonian field theories on manifolds with boundaries. This work is a geometric fulfillment of Fock's characterization of field theories as it appears in recent work by Cattaneo, Mnev and Reshetikhin [Ca14]. This framework leads to a true geometric understanding of conventional choices for boundary conditions. For example, the boundary condition that the pull-back of the 1-form on the cotangent space of fields at the boundary vanish, i.e. \pi * \alpha = 0 , is shown to be a consequence of our finding that the boundary fields of the theory lie in the 0-level set of the moment map of the gauge group of the theory. It is also shown that the natural way to interpret Euler-Lagrange equations as an evolution system near the boundary is as a presymplectic system in an extended phase space containing the natural configuration and momenta fields at the boundary together with extra degrees of freedom corresponding to the transversal components at the boundary of the momenta fields of the theory. The consistency conditions at the boundary are analyzed and the reduced phase space of the system is determined to be a symplectic manifold with a distinguished isotropic submanifold corresponding to the boundary data of the solutions of Euler-Lagrange equations. This setting makes it possible to define well-posed boundary conditions, and provides the adequate setting for the canonical quantization of the system. The notions of the theory will be tested against three significant examples: scalar fields, Poisson\sigma-model and Yang-Mills theories.