Generalised Cluster Adjacency for Cosmology

TL;DR

This paper explores the algebraic structure of wavefunction coefficients in de Sitter cosmology, introducing a generalized cluster adjacency property that constrains the symbol bootstrap for massless scalar theories.

Contribution

It introduces the ordered single cluster condition, a stronger form of cluster adjacency, and extends cluster-like structures to tree graphs for improved bootstrap methods.

Findings

Wavefunction coefficients obey a generalized cluster adjacency in de Sitter cosmology.

The ordered single cluster condition provides new constraints for the symbol bootstrap.

Tree graphs exhibit a cluster-like structure using tubes and tubings, aiding bootstrap approaches.

Abstract

In this paper we study the cluster algebraic properties of wavefunction coefficients for massless scalar theories in de Sitter cosmology. We show that the symbol of the wavefunction coefficient of the $n$-site path graph $P_n$ obeys a generalisation of cluster adjacency, where all letters in a given word belong to the same cluster of an $A_{2n-3}$ algebra, with certain additional constraints on the order of the letters. We call this property the ordered single cluster condition, and provide its physical interpretation. This condition is stronger than the usual cluster adjacency obeyed by neighbouring letters, and imposes stronger constraints for the symbol bootstrap. We also show how any tree graph satisfies a cluster-like structure in terms of tubes and tubings on the underlying graph, which allows for a similar bootstrap approach.