Wald-Zoupas prescription with (soft) anomalies

TL;DR

This paper demonstrates that the Wald–Zoupas prescription for gravitational charges extends to settings with field-dependent diffeomorphisms and soft anomalies, provided these anomalies satisfy a precise covariance relation. It shows that, when a corner term solving a specific differential equation exists, WZ charges can be recast as improved Noether charges with anomaly-free boundary data, offering an a priori construction of boundary Lagrangians and corner terms. The four detailed examples, especially future null infinity, reveal that soft anomalies correspond to memory terms in BMS flux-balance laws, clarifying why WZ yields the standard BMS charges despite anomalous transformations. Overall, the work unifies disparate strands of the covariant phase space literature by relating WZ charges to boundary-improved Noether charges and by highlighting the physical role of soft anomalies in gravitational surface charges.

Abstract

We show that the Wald-Zoupas prescription for gravitational charges is valid in the presence of anomalies and field-dependent diffeomorphism, but only if these are related to one another in a specific way. The geometric interpretation of the allowed anomalies is exposed looking at the example of BMS symmetries: They correspond to soft terms in the charges. We determine if the Wald-Zoupas prescription coincides with an improved Noether charge. The necessary condition is a certain differential equation, and when it is satisfied, the boundary Lagrangian of the resulting improved Noether charge contains in general a non-trivial corner term that can be identified a priori from a condition of anomaly-freeness. Our results explain why the Wald-Zoupas prescription works in spite of the anomalous behaviour of BMS transformations, and should be helpful to relate different branches of the literature on surface charges.