Cluster Bootstrap for Cosmological Correlators

TL;DR

This paper uncovers the cluster algebra structure of cosmological wavefunction coefficients in de Sitter space, linking graph combinatorics to cluster variables, and uses this to bootstrap explicit symbols for low-order cases.

Contribution

It establishes a novel connection between graph tubings, polygon triangulations, and cluster algebras in cosmological correlators, enabling bootstrap of de Sitter symbols.

Findings

Cluster variables form the symbol alphabet for cosmological correlators.

Cluster adjacency properties hold for wavefunction coefficients in de Sitter space.

De Sitter symbols for n ≤ 4 are uniquely fixed by physical constraints.

Abstract

We show that cosmological wavefunction coefficients associated with $n$-site chain and loop graphs for a cubic scalar theory in de Sitter spacetime have symbol alphabets given by subsets of $A_{2n{-}2}$ and $B_{2n{-}1}$ cluster variables, respectively, and satisfy the associated cluster adjacency properties. The key step in proving this is identifying a precise connection between graph "tubings" that appear in the kinematic flow equation and polygon "triangulations" that encode the combinatorics of cluster compatibility. Our results imply that cosmological wavefunction coefficients in a general power-law FRW cosmology satisfy cluster adjacency to all orders in the $\epsilon$ expansion. We use this information as bootstrap input to show that de Sitter symbols for $n \leq 4$ are uniquely determined by simple physical constraints.