Cluster algebras for cosmological correlators

TL;DR

The paper investigates how the singularity structure of cosmological correlators in a conformally coupled scalar theory can be organized by cluster algebras. By analyzing the differential equations governing these correlators, the authors map the symbol letters to $A_n$-type algebras, showing that tree-level ladder n-site correlators correspond to $A_{2(n-1)}$ with specific letter exclusions, and that one-loop bubble correlators decompose into a union of two $A_3$ algebras. They provide explicit transformed alphabets for low-site cases and embed the letters geometrically via polygon triangulations, offering a clear polygon-cluster correspondence. The results suggest a robust bootstrap framework for analytic cosmological correlators and point toward further connections with cluster adjacency and positive geometry.

Abstract

In this paper, we explore the cluster algebras for symbol letters or singularities of cosmological correlators in a conformally coupled scalar field theory. We show that the symbol letters for tree-level n-site ladder cosmological correlators are governed by A_{2(n-1)} cluster algebras. Additionally, we demonstrate that the symbol letters for one-loop bubble cosmological correlator are an union of two A_3 cluster algebras. The algebras relations of letters will provide an important tool to bootstrap analytic cosmological correlators.

Figures (10)

• Figure 1: An example of a quiver mutation.
• Figure 2: Triangulation flip of a diagonal in quadrilateral.
• Figure 3: Realization model of $A_2$ cluster algebras with a 5-gon, where the dots represent mutable vertices, while the squares denote frozen variables
• Figure 4: Tree-level n-site ladder cosmological correlators.
• Figure 5: Quadrilateral associated to the letters of two-sites correlators.