More on Bulk Local State Reconstruction in Flat/Carr CFT

TL;DR

This paper advances the understanding of bulk local state reconstruction in flat holography across dimensions, clarifying the flat limit process and introducing a dual basis to ensure smoothness and correct Green's functions.

Contribution

It introduces a dual basis for flat holography, clarifies the flat limit in higher dimensions, and establishes a unified algebraic framework for bulk reconstruction.

Findings

Resolved the scaling mismatch in 3D flat basis using a dual basis.

Constructed explicit bulk local states in higher dimensions.

Revealed the non-uniform flat limit and recovered the massive propagator.

Abstract

We revisit and extend the construction of bulk local states in flat holography, focusing on the induced representation obtained from the flat limit of the AdS highest-weight conditions. In three dimensions we clarify the scaling mismatch between bra and ket states in the flat basis and resolve it by introducing a dual basis, which yields a smooth flat limit and reproduces the correct Green's function. For higher dimensions we construct bulk local states explicitly, both in the momentum basis and in an alternative tilde basis. The flat limit of the AdS$_{d+1}$ construction is shown to be non-uniform in the descendant level and the Riemann-sum treatment over the scaling window $n\sim l$ converts the discrete descendant expansion into the continuum momentum representation, recovering the massive propagator. The tilde basis generalizes seamlessly to any dimension and is related to the three-dimensional flat basis by a sign factor. These results establish the induced representation as the correct algebraic foundation for bulk reconstruction in flat holography and provide a unified framework valid for arbitrary dimension.