  In the context of the quest for a holographic formulation of quantum gravity,
we investigate the basic boundary theory structure for loop quantum gravity. In
3+1 space-time dimensions, the boundary theory lives on the 2+1-dimensional
time-like boundary and is supposed to describe the time evolution of the edge
modes living on the 2-dimensional boundary of space, i.e. the space-time
corner. Focusing on "electric" excitations -- quanta of area -- living on the
corner, we formulate their dynamics in terms of classical spinor variables and
we show that the coupling constants of a polynomial Hamiltonian can be
understood as the components of a background boundary 2+1-metric. This leads to
a deeper conjecture of a correspondence between boundary Hamiltonian and
boundary metric states. We further show that one can reformulate the quanta of
area data in terms of a SL(2,C) connection, transporting the spinors on the
boundary surface and whose SU(2) component would define "magnetic" excitations
(tangential Ashtekar-Barbero connection), thereby opening the door to writing
the loop quantum gravity boundary dynamics as a 2+1-dimensional SL(2,C) gauge
theory.
