Gravitational $SL(2,\mathbb{R})$ Algebra on the Light Cone

TL;DR

The paper addresses how edge modes and radiative data on a null boundary are organized within the parity-odd Palatini--Holst gravity, and how the Barbero--Immirzi parameter $\gamma$ deforms the boundary $SL(2,\mathbb{R})$ symmetry into a $U(1)$ structure. It constructs a covariant radiative phase space for the bulk plus null boundary, introduces an auxiliary $SL(2,\mathbb{R})$ phase space to compute Dirac brackets, and performs a detailed Dirac analysis of first- and second-class constraints. The key results include the identification of two radiative boundary degrees of freedom, the derivation of the boundary pre-symplectic potential and its gauge symmetries, and the explicit form of Dirac observables such as the dressed $SL(2,\mathbb{R})$ holonomies and relational area measures along null generators. The work provides a concrete framework for edge-mode quantization and connects to loop quantum gravity via the $\gamma$-dependent U(1) structure and area quantization, while outlining viable truncations to render the theory tractable for quantum analysis.

Abstract

In a region with a boundary, the gravitational phase space consists of radiative modes in the interior and edge modes at the boundary. Such edge modes are necessary to explain how the region couples to its environment. In this paper, we characterise the edge modes and radiative modes on a null surface for the tetradic Palatini--Holst action. Our starting point is the definition of the action and its boundary terms. We choose the least restrictive boundary conditions possible. The fixed boundary data consists of the radiative modes alone (two degrees of freedom per point). All other boundary fields are dynamical. We introduce the covariant phase space and explain how the Holst term alters the boundary symmetries. To infer the Poisson brackets among Dirac observables, we define an auxiliary phase space, where the $SL(2\mathbb{R})$ symmetries of the boundary fields is manifest. We identify the gauge generators and second-class constraints that remove the auxiliary variables. All gauge generators are at most quadratic in the fundamental $SL(2,\mathbb{R})$ variables on phase space. We compute the Dirac bracket and identify the Dirac observables on the light cone. Finally, we discuss various truncations to quantise the system in an effective way.