A note on the Hamiltonian structure of transgression forms

Abstract

By incorporating two gauge connections, transgression forms provide a generalization of Chern-Simons actions that are genuinely gauge-invariant on bounded manifolds. In this work, we show that, when defined on a manifold with a boundary, the Hamiltonian formulation of a transgression field theory can be consistently carried out without the need to implement regularizing boundary terms at the level of first-class constraints. By considering boundary variations of the relevant functionals in the Poisson brackets, the surface integral in the very definition of a transgression action can be translated into boundary contributions in the generators of gauge transformations and diffeomorphisms. This prescription systematically leads to the corresponding surface charges of the theory, reducing to the general expression for conserved charges in (higher-dimensional) Chern-Simons theories when one of the gauge connections in the transgression form is set to zero.

Figures (2)

• Figure 1: Cobordant manifolds. Two manifolds $\mathcal{M}$ and $\bar{\mathcal{M}}$ having a common boundary $\partial \mathcal{M}=\partial\bar{\mathcal{M}}$.
• Figure 2: Gauged WZW case. A single manifold $\mathcal{M}$ where two connections $A$ and $A^{h}$ are defined, belonging to their respective spaces $G_{p}$ and $G'_{p}$ at $p\in\mathcal{M}$. $A$ and $A^{h}$ are connected by a gauge transformation defined by an element $h\in G$, where $G$ is the strucutre group.