  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.
