Edge modes of gravity. Part III. Corner simplicity constraints
Laurent Freidel, Marc Geiller, Daniele Pranzetti
TL;DR
This work develops a rigorous corner-phase-space framework for tetrad gravity, embedding the BF formulation into a Poincaré–Heisenberg precursor and then imposing kinematical and simplicity constraints to recover BF/Einstein–Cartan–Holst gravity at the corner. It identifies an explicit split of the corner simplicity constraints into first- and second-class parts, constructs Dirac observables including an SL(2,C) Lorentz sector and an SL(2,R) parallel sector, and shows that the corner area is a Poincaré spin Casimir with the Barbero–Immirzi parameter playing the role of a mass. The paper develops both continuous and discretized representations of the corner symmetry algebra, connects to twisted geometries, and lays groundwork for a covariant quantization of quantum geometry through edge modes, with future work to fully realize the Corner-Mode-IV program. Altogether, it resolves key tensions between Lorentz invariance and area discreteness in LQG and provides a locally holographic, corner-based route toward quantum gravity.
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 $\mathfrak{sl}(2,\mathbb{C})$ subalgebra of Poincaré, and the components of the tangential corner metric satisfying an $\mathfrak{sl}(2,\mathbb{R})$ 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.