Revisiting 3D Flat Holography: Causality Structure and Modular flow

Abstract: Flat space holography is an open and hard problem existing several different approaches, which may finally turn out to be consistent with each other, in the literature to tackle it. Focusing on how bulk emergent spacetime is encoded in quantum information of null boundaries, we choose a specific toy model called the flat$_3$/BMSFT model, which conjectures the duality between boundary BMS$_3$ invariant field theory and bulk quantum gravity in 3D asymptotic flat spacetimes (AFS), to explore. Aiming to find an entanglement wedge like quantity for single interval and a connected entanglement wedge for multi-intervals in flat$_3$/BMSFT model, we explore the bulk causality structures related to the holographic swing surface proposal through both boundary and bulk local modular flow, make a corresponding decomposition of the global Minkowski spacetime and look at the entanglement phase transition. As a byproduct, we solve the problem about the existence of partial entanglement entropy (PEE) correspondence in this model which is a bit nontrivial due to the unusual behavior of boundary modular flow in BMS$_3$ field theory. Among the literature considering quantum information aspects of flat$_3$/BMSFT model, there are several substantial, unusual but overlooked phenomena which need to be emphasized and revisited to gain more deserved attention. Thus another motivation of this paper is to find where these unusual phenomena come from, and physically show in a manifest way what they may imply. After reading we hope readers can feel sincerely what we present about the above mentioned second aim is more valuable than the mathematical results in the present paper.