Loop Quantum Gravity Boundary Dynamics and SL(2,C) Gauge Theory

TL;DR

This paper explores the boundary dynamics of loop quantum gravity in 3+1 dimensions, proposing a reformulation as an SL(2,C) gauge theory that links boundary Hamiltonians to metric states and introduces new boundary excitations.

Contribution

It introduces a novel boundary theory framework for loop quantum gravity using classical spinors and reformulates boundary data as an SL(2,C) gauge connection, connecting boundary Hamiltonians to metric states.

Findings

Boundary dynamics described by classical spinor variables.

Coupling constants relate to boundary 2+1-metric components.

Boundary data reformulated as SL(2,C) gauge connection.

Abstract

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.