Finite Charges from the Bulk Action
Robert McNees, Céline Zwikel
TL;DR
The paper develops a systematic method to obtain finite, integrable charges in the covariant phase space by identifying corner terms directly from the variation of the bulk action. By shifting the presymplectic potential to absorb corner contributions, they construct a renormalized, $r$-independent codimension-2 form that yields well-defined charges without relying on particular boundary Lagrangians or gauges. Applying the procedure to 2d dilaton gravity and 3d Einstein gravity in Bondi-type gauges, they derive explicit charges, show their integrability (often with a Heisenberg-like algebra), and establish corresponding variational principles via holographic renormalization, including corner terms on boundaries with corners. The results illuminate the role of off-shell (partially on-shell) configurations and weakly vanishing Noether currents in charge construction, with implications for holography and flat-space limits. While demonstrated in low dimensions, the framework is presented as general and potentially extensible to higher dimensions and additional gauge fields, offering a unified view of finite charges in diffeomorphism-invariant theories.
Abstract
Constructing charges in the covariant phase space formalism often leads to formally divergent expressions, even when the fields satisfy physically acceptable fall-off conditions. These expressions can be rendered finite by corner ambiguities in the definition of the presymplectic potential, which in some cases may be motivated by arguments involving boundary Lagrangians. We show that the necessary corner terms are already present in the variation of the bulk action and can be extracted in a straightforward way. Once these corner terms are included in the presymplectic potential, charges derived from an associated codimension-2 form are automatically finite. We illustrate the procedure with examples in two and three dimensions, working in Bondi gauge and obtaining integrable charges. As a by-product, actions are derived for these theories that admit a well-defined variational principle when the fields satisfy boundary conditions on a timelike surface with corners. An interesting feature of our analysis is that the fields are not required to be fully on-shell.