Hamiltonian analysis of a topological theory in the presence of boundaries

Abstract

We perform the canonical Hamiltonian analysis of a topological gauge theory, that can be seen both as a theory defined on a four dimensional spacetime region with boundaries --the bulk theory--, or as a theory defined on the boundary of the region --the boundary theory--. In our case the bulk theory is given by the 4-dimensional $U(1)$ Pontryagin action and the boundary one is defined by the $U(1)$ Chern-Simons action. We analyse the conditions that need to be imposed on the bulk theory so that the total Hamiltonian, smeared constraints and generators of gauge transformations be well defined (differentiable) for generic boundary conditions. We pay special attention to the interplay between the constraints and boundary conditions in the bulk theory on the one side, and the constraints in the boundary theory, on the other side. We illustrate how both theories are equivalent, despite the different canonical variables and constraint structure, by explicitly showing that they both have the same symmetries, degrees of freedom and observables.

arXiv comment15 pages, 1 Table. Version to be published in IJMPD

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Cited by in corpus (7)

• Dirac's algorithm in the presence of boundaries: a practical guide to a geometric approach
• Canonical analysis of field theories in the presence of boundaries: Maxwell+Pontryagin
• Canonical aspects of pregeometric vector-based first order gauge theory
• Weakly Isolated Horizons: $3+1$ decomposition and canonical formulations in self-dual variables
• On covariant and canonical Hamiltonian formalisms for gauge theories
• Constructing the theory at the boundary, its dynamics and degrees of freedom
• The Hamilton-Jacobi characteristic equations for topological invariants: Pontryagin and Euler classes