Conformal boundary conditions, loop gravity and the continuum

TL;DR

This work shows that in three-dimensional Euclidean gravity with zero cosmological constant, the discrete spectra of boundary geometric observables from loop quantum gravity can be obtained from quantising a conformal boundary field theory in the continuum, avoiding spin networks. By formulating a bulk-plus-boundary system with conformal boundary conditions encoded by an SU(2) boundary spinor, the authors derive a minimal-surface (holomorphic) boundary dynamics that yields a boundary CFT with vanishing central charge; the boundary length operator emerges as a sum of harmonic-oscillator quanta, producing a discrete spectrum in the boundary theory. The paper develops a continuum boundary Hilbert space, introduces coherent states and a quasi-local partition function, and relates the boundary CFT to LQG by embedding discrete spin-network data into the continuum boundary Fock space, while highlighting conceptual and technical connections to group field theory. Overall, it provides a continuum formulation that reproduces discrete geometric features, clarifies the role of boundary edge modes, and opens avenues for comparing with spinfoam and GFT approaches using conformal boundary data. The results emphasize the primacy of boundary dynamics in 3D gravity and suggest a path to understanding quasi-local holography and geometric quantisation without triangulations.

Abstract

In this paper, we will make an attempt to clarify the relation between three-dimensional euclidean loop quantum gravity with vanishing cosmological constant and quantum field theory in the continuum. We will argue, in particular, that in three spacetime dimensions the discrete spectra for the geometric boundary observables that we find in loop quantum gravity can be understood from the quantisation of a conformal boundary field theory in the continuum without ever introducing spin networks or triangulations of space. At a technical level, the starting point is the Hamiltonian formalism for general relativity in regions with boundaries at finite distance. At these finite boundaries, we choose specific conformal boundary conditions (the boundary is a minimal surface) that are derived from a boundary field theory for an SU(2) boundary spinor, which is minimally coupled to the spin connection in the bulk. The resulting boundary equations of motion define a conformal field theory with vanishing central charge. We will quantise this boundary field theory and show that the length of a one-dimensional cross section of the boundary has a discrete spectrum. In addition, we will introduce a new class of coherent states, study the quasi-local observables that generate the quasi-local Virasoro algebra and discuss some strategies to evaluate the partition function of the theory.