3d gravity in Bondi-Weyl gauge: charges, corners, and integrability

Abstract

We introduce a new gauge and solution space for three-dimensional gravity. As its name Bondi-Weyl suggests, it leads to non-trivial Weyl charges, and uses Bondi-like coordinates to allow for an arbitrary cosmological constant and therefore spacetimes which are asymptotically locally (A)dS or flat. We explain how integrability requires a choice of integrable slicing and also the introduction of a corner term. After discussing the holographic renormalization of the action and of the symplectic potential, we show that the charges are finite, symplectic and integrable, yet not conserved. We find four towers of charges forming an algebroid given by $$ \mathfrak{vir}\oplus \mathfrak{vir}\oplus $$ vir ⊕ vir ⊕ Heisenberg with three central extensions, where the base space is parametrized by the retarded time. These four charges generate diffeomorphisms of the boundary cylinder, Weyl rescalings of the boundary metric, and radial translations. We perform this study both in metric and triad variables, and use the triad to explain the covariant origin of the corner terms needed for renormalization and integrability.

Null boundary phase space: slicings, news & memory

“…The conjecture put forward in [1] and verified for some examples in [16][17][18][19]30] states that there exist phase space slicings in which there are no fake news, and the non-integrable part of the surface charges is determined entirely by genuine news. In other words, the conjecture posits that there always exists at least one genuine slicing.…”

“…therein. For D < 4 such an analysis was carried through in [16], see also [17][18][19]. For generic D there are numerous earlier constructions, see e.g.…”

“…In practice, however, it can also happen that the surface charges are not integrable in the absence of any physical fluxes. As mentioned in [1] and made explicit in [16][17][18][19]30], integrability of the surface charges depends on the slicing used to describe the boundary phase space. We are going to be more explicit about what we mean by 'slicing' in the body of our paper.…”

“…An important aspect discussed, e.g., in [16][17][18][19] is that the integrability of the charges and the presence or absence of fluxes do depend on the slicing. In the following, to shed new light on this issue, we discuss two classes of slicings.…”

“…Inevitably, we need to consider another mode, switched off by our assumptions in section 2, namely an O(1) term in V in the expansion (2.4), which relaxes the condition that our boundary N is null. See [19] for the D = 3 example. This generalization adds an extra freedom and a corresponding new charge.…”