The Quantum Gravity Disk: Discrete Current Algebra

Abstract

We study the quantization of the corner symmetry algebra of 3d gravity associated with 1d spatial boundaries. We first recall that in the continuum, this symmetry algebra is given by the central extension of the Poincaré loop algebra. At the quantum level, we construct a discrete current algebra with a DSU(2) quantum symmetry group that depends on an integer N . This algebra satisfies two fundamental properties: First it is compatible with the quantum space-time picture given by the Ponzano-Regge state-sum model, which provides a path integral amplitudes for 3d loop quantum gravity. We then show that we recover in the N → ∞ limit the central extension of the Poincaré current algebra. The number of boundary edges defines a discreteness parameter N which counts the number of flux lines attached to the boundary. We analyse the refinement, coarse-graining and fusion processes as N changes. Identifying a discrete current algebra on quantum boundaries is an important step towards understanding how conformal field theories arise on spatial boundaries in loop quantum gravity. It also shows how an asymptotic BMS symmetry group could appear from the continuum limit of 3d quantum gravity. Contents B. Commutators of the tentative discrete Virasoro operators Dn

The Ponzano–Regge cylinder and propagator for 3d quantum gravity

“…Beyond the exploration of possible extensions of the present results in the realm of 3d quantum gravity (for example, studying cylindric transition amplitudes for arbitrary spin superpositions as in [32,[38][39][40]), our work further opens possible doors to applications beyond the Ponzano-Regge model:…”

Three-dimensional de Sitter horizon thermodynamics

“…This has been explored, for example, in [32,56]. 7 The operators : e ±i ζ : correspond to insertions of fractional charge ±1/3 inside the spatial disk, and are primaries under the N = 2 superconformal algebra.…”

“…This limit might also suggest we treat the de Sitter horizon as a boundary[3][4][5][6][7], although a clear framework for how to do so remains unclear.…”