Generating functional for gravitational null initial data

TL;DR

The paper constructs a boundary field theory on a null three-manifold that encodes the vacuum Einstein constraint equations on a null hypersurface, achieved by doubling the boundary content across an interface and coupling a kinetic term for edge spinors to a dressed Chern–Simons action. By imposing gluing, torsionlessness, and extrinsic matching via Lagrange multipliers, the boundary dynamics reproduce the null-boundary GR constraints while keeping gravitational radiation as boundary data through external Weyl components. It then shows how transition amplitudes for edge states, parameterized by the boundary flux class $[m_a]$, can be glued to yield quasi-local bulk amplitudes, offering a holographic-like, light-front realization of gravity with edge modes living on the boundary and flux data encoding bulk information. This framework provides a pathway to define bulk quantum states via auxiliary boundary theories and to study gravitational transition amplitudes in a gauge-invariant, quasi-local setting that mirrors holographic principles at finite boundaries.

Abstract

A field theory on a three-dimensional manifold is introduced, whose field equations are the constraint equations for general relativity on a three-dimensional null hypersurface. The underlying boundary action consists of two copies of the dressed Chern-Simons term for self-dual Ashtekar variables, a kinetic term for the null flag at the boundary plus additional junction conditions for the spin coefficients across the interface. In fact, there is a doubling of the field content, because the null hypersurface will be considered as an internal boundary between two adjacent slabs of spacetime. The paper concludes with a proposal for a construction of the gravitational transition amplitudes in the bulk via the auxiliary boundary field theory alone, namely by gluing amplitudes for edge states across two-dimensional corners, thus providing a proposal for a quasi-local realisation of the holographic principle at the light front.