Conformal boundary conditions, loop gravity and the continuum

TL;DR

This paper links 3D Euclidean loop quantum gravity with boundary conformal field theory, showing how discrete geometric spectra emerge from continuum boundary quantization without spin networks.

Contribution

It demonstrates that boundary conformal field theory can reproduce discrete geometric spectra in 3D loop quantum gravity, avoiding the need for triangulations.

Findings

Discrete spectra for boundary length observables derived from boundary CFT

Quantisation of boundary SU(2) spinor field yields a conformal theory with zero central charge

Introduction of new coherent states and analysis of boundary Virasoro algebra

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.