Edge modes of gravity. Part II. Corner metric and Lorentz charges

Dual gravitational charges have been recently computed from the Holst term in tetrad variables using covariant phase space methods. We highlight that they originate from an exact 3-form in the tetrad symplectic potential that has no analogue in metric variables. Hence there exists a choice of the tetrad symplectic potential that sets the dual charges to zero. This observation relies on the ambiguity of the covariant phase space methods. To shed more light on the dual contributions, we use the Kosmann variation to compute (quasi-local) Hamiltonian charges for arbitrary diffeomorphisms. We obtain a formula that illustrates comprehensively why the dual contribution to the Hamiltonian charges: (i) vanishes for exact isometries and asymptotic symmetries at spatial infinity; (ii) persists for asymptotic symmetries at future null infinity, in addition to the usual BMS contribution. Finally, we point out that dual gravitational charges can be equally derived using the Barnich-Brandt prescription based on cohomological methods, and that the same considerations on asymptotic symmetries apply.

Section: Jhep12(2020)079mentioning

confidence: 99%

A note on dual gravitational charges

Oliveri

Speziale

2020

“…For related works in other formalisms see e.g. [24,[67][68][69][70][71][72][73][74][75][76][77][78][79][80][81]. Note that it is expected that different formalisms lead to different symmetry algebras [77].…”

Section: Introductionmentioning

confidence: 99%

Conservation and integrability in lower-dimensional gravity

Ruzziconi

Zwikel

2021

113

We address the questions of conservation and integrability of the charges in two and three-dimensional gravity theories at infinity. The analysis is performed in a framework that allows us to treat simultaneously asymptotically locally AdS and asymptotically locally flat spacetimes. In two dimensions, we start from a general class of models that includes JT and CGHS dilaton gravity theories, while in three dimensions, we work in Einstein gravity. In both cases, we construct the phase space and renormalize the divergences arising in the symplectic structure through a holographic renormalization procedure. We show that the charge expressions are generically finite, not conserved but can be made integrable by a field-dependent redefinition of the asymptotic symmetry parameters.

“…We have recently proposed in [1,2] a new local holographic perspective on quantum gravity. The aim is to study the notion of corner symmetry algebra in gravity, with the expectation that understanding its representation theory will reveal universal features of quantum gravity [3].…”

Section: Introductionmentioning

confidence: 99%

“…In the tetrad formulation of gravity with non-vanishing Barbero-Immirzi parameter, the corner symplectic structure contains the internal normal to the foliation n I as a dynamical variable as well as the corner coframe field, and in this case the symmetry group contains a factor G = SL(2, C) × SL(2, R) . We have shown in [2] that the corner sl(2, R) algebra, which is generated by the tangential corner metric components, is at the origin of the discreteness of the corner area element. This illustrates the non-trivial semi-classical physical information encoded in the corner symplectic structure.…”

Section: Introductionmentioning

confidence: 99%

Edge modes of gravity. Part III. Corner simplicity constraints

Freidel

Geiller

Pranzetti

2021

Self Cite

103

In the tetrad formulation of gravity, the so-called simplicity constraints play a central role. They appear in the Hamiltonian analysis of the theory, and in the Lagrangian path integral when constructing the gravity partition function from topological BF theory. We develop here a systematic analysis of the corner symplectic structure encoding the symmetry algebra of gravity, and perform a thorough analysis of the simplicity constraints. Starting from a precursor phase space with Poincaré and Heisenberg symmetry, we obtain the corner phase space of BF theory by imposing kinematical constraints. This amounts to fixing the Heisenberg frame with a choice of position and spin operators. The simplicity constraints then further reduce the Poincaré symmetry of the BF phase space to a Lorentz subalgebra. This picture provides a particle-like description of (quantum) geometry: the internal normal plays the role of the four-momentum, the Barbero-Immirzi parameter that of the mass, the flux that of a relativistic position, and the frame that of a spin harmonic oscillator. Moreover, we show that the corner area element corresponds to the Poincaré spin Casimir. We achieve this central result by properly splitting, in the continuum, the corner simplicity constraints into first and second class parts. We construct the complete set of Dirac observables, which includes the generators of the local $$ \mathfrak{sl}\left(2,\mathrm{\mathbb{C}}\right) $$ sl 2 ℂ subalgebra of Poincaré, and the components of the tangential corner metric satisfying an $$ \mathfrak{sl}\left(2,\mathrm{\mathbb{R}}\right) $$ sl 2 ℝ algebra. We then present a preliminary analysis of the covariant and continuous irreducible representations of the infinite-dimensional corner algebra. Moreover, as an alternative path to quantization, we also introduce a regularization of the corner algebra and interpret this discrete setting in terms of an extended notion of twisted geometries.