Edge modes of gravity. Part III. Corner simplicity constraints

Abstract

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.

Chiral massive news: null boundary symmetries in topologically massive gravity

“…interesting algebraic question, one may classify all such slicings in which these two features are present. On the other hand, the fact that such v-independent slicings exist, dovetails with the "corner symmetry" picture [6][7][8]. Namely, one may take the corner to be at constant r, v codimension 2 surfaces.…”

“…In a series of papers[6][7][8] a closely related notion of corner symmetries have been developed and studied.…”

Charge algebra in Al(A)dSn spacetimes

“…As discussed in [35], the form of the Λ-BMS 4 charge algebra may be affected by the addition of corner terms [120][121][122][123] at the level of the variational principle (3.12) that are necessary to take the flat limit at the level of the symplectic structure.…”

Conservation and integrability in lower-dimensional gravity

“…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].…”