From AdS to dS Exchanges: Spectral Representation, Mellin Amplitudes and Crossing

TL;DR

This paper establishes a simple relation between AdS and dS tree-level exchanges, enabling the transfer of techniques like Mellin amplitudes and spectral representations from AdS to dS boundary correlators, and deriving conformal block decompositions.

Contribution

It introduces a direct relation between AdS and dS exchanges, allowing the import of AdS techniques to dS and deriving spectral and Mellin representations for dS exchanges.

Findings

Derived a relation between AdS and dS exchanges.

Defined Mellin amplitudes and spectral representations for dS.

Obtained conformal block decompositions in dS from AdS counterparts.

Abstract

We present a simple general relation between tree-level exchanges in AdS and dS. This relation allows to directly import techniques and results for AdS Witten diagrams, both in position and momentum space, to boundary correlation functions in dS. In this work we apply this relation to define Mellin amplitudes and a spectral representation for exchanges in dS. We also derive the conformal block decomposition of a dS exchange, both in the direct and crossed channels, from their AdS counterparts. The relation between AdS and dS exchanges itself is derived using a recently introduced Mellin-Barnes representation for boundary correlators in momentum space, where (A)dS exchanges are straightforwardly fixed by a combination of factorisation, conformal symmetry and boundary conditions.