Gravitational edge modes: from Kac–Moody charges to Poincaré networks

Abstract

We revisit the canonical framework for general relativity in its connectionvierbein formulation, recasting the Gauss law, the Bianchi identity and the space diffeomorphism bulk constraints as conservation laws for boundary surface charges, respectively electric, magnetic and momentum charges. Partitioning the space manifold into 3D regions glued together through their interfaces, we focus on a single domain and its punctured 2D boundary. The punctures carry a ladder of Kac-Moody edge modes, whose 0-modes represent the electric and momentum charges while the higher modes describe the stringy vibration modes of the 1D-boundary around each puncture. In particular, this allows to identify missing observables in the discretization scheme used in loop quantum gravity and leads to an enhanced theory upgrading spin networks to tube networks carrying Virasoro representations. In the limit where the tubes are contracted to 1D links and the string modes neglected, we do not just recover loop quantum gravity but obtain a more general structure: Poincaré charge networks, which carry a representation of the 3D diffeomorphism boundary charges on top of the SU(2) fluxes and gauge transformations.

Kinematical gravitational charge algebra

“…We prove that these kinematical charges generate a local Poincaré ISU(2) symmetry algebra. This gives strong support to the recent proposal of Poincaré charge networks as a new realm for discretized general relativity [1]. *…”

“…In four and higher space-time dimensions, edge modes have been argued to be essential to understanding black hole entropy [13][14][15][16]. Finally they are now understood to be a key ingredient in the holographic nature of gravity [17][18][19][20] and they provide a new understanding in the quantization of geometric observables [1,21,22].…”

“…In [1], relying on the assumption that curvature on the boundary is distributional (i.e. it vanishes on the boundary, except at the location of a given set of points or punctures), we exploited the local holographic nature of gravity to put forward an extension of the kinematical quantum degrees of freedom given by the spin networks in Loop Quantum Gravity.…”

“…it vanishes on the boundary, except at the location of a given set of points or punctures), we exploited the local holographic nature of gravity to put forward an extension of the kinematical quantum degrees of freedom given by the spin networks in Loop Quantum Gravity. The analysis of [1], motivated by a realization of quantum gravity as dynamical networks of quantum edge modes, led to a novel construction of tubular networks dressed by representations of the Euclidean iso(3) algebra generated by the fluxes and momentum operators (plus the additional higher mode quantum numbers). This is done by gluing together 3D regions bounded by 2D surfaces with punctures where the edge modes live and which are presumed to carry all the dynamical information of the 3D bulk geometry.…”

Spin networks and cosmic strings in 3+1 dimensions

“…Note that if M = 0 and S = 0, the particle is indistinguishable from flat spacetime 28 . 28 It is interesting to compare this to the 2+1D case, which is discussed in [29,30,31,32,33]. There we have point particles instead of strings.…”