# Holography in de Sitter and anti-de Sitter Spaces and Gel'fand Graev Radon transform

## TL;DR

This paper develops holographic reconstruction formulas for de Sitter and anti-de Sitter spaces using the Gel'fand Graev Radon transform, extending previous work and deriving explicit Wightman functions.

## Contribution

It introduces a new inverse Radon transform approach for bulk reconstruction in de Sitter and anti-de Sitter spaces, valid in odd dimensions and applicable to black hole geometries.

## Key findings

- Reconstruction formulas are valid in odd dimensions for both spaces.
- Explicit analytical Wightman functions are derived for de Sitter space.
- The formulas are applicable to the BTZ black hole case.

## Abstract

Bulk reconstruction formulas similar to HKLL are obtained for de Sitter and anti-de Sitter spaces as the inverse Gel'fand Graev Radon transform. While these generalize our previous result on the Euclidean anti-de Sitter space, their validity in here is restricted only to odd dimensions in both instances. The exact Wightman function for the de Sitter space is then derived analytically. The GGR transform fixes the coefficient of the Wightman function. For the anti-de Sitter space it is shown that a reconstruction formula exists for the case of time-like boundary as well. The restriction on the domain of integration on the boundary is derived. As a special case, we point out that the formula is valid for the BTZ black hole as well.