Cluster algebras for cosmological correlators

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.

💡 Research Summary

The paper investigates the appearance of cluster algebras in the symbol letters (or singularities) of cosmological correlators arising from a conformally coupled scalar field theory. After a brief review of cluster algebras, seeds, mutations, and the special role of A‑type algebras, the authors focus on two families of correlators: tree‑level ladder diagrams with n external sites and the one‑loop bubble diagram.

For the tree‑level ladder correlators, the authors first list the alphabet of logarithmic letters that appear in the ε‑factorised differential equations governing the correlators. In the simplest two‑site case the alphabet consists of five linear combinations of the kinematic variables X_i and Y_i. By introducing a change of variables
 z₁ = 2 Y₁ / (X₁ − Y₁), z₂ = (X₂ + Y₁) / (X₁ − Y₁),
the five letters are mapped to the set {z₁, 1+z₁, z₂, 1+z₂, z₁−z₂}, which is precisely the set of A₂ cluster coordinates. The same procedure is applied to three‑site and four‑site ladders. After suitable re‑definitions of the kinematic variables, the resulting alphabets match the A₄ and A₆ cluster algebras, respectively, up to a few missing differences such as z₂−z₃. The pattern generalises: an n‑site ladder correlator yields an alphabet that is (almost) the A_{2(n−1)} cluster algebra. The missing letters are interpreted as consequences of dimensional constraints or redundancies introduced by the variable change.

The authors then give a geometric interpretation of these letters. Using Ptolemy’s theorem for cyclic quadrilaterals, they show that the algebraic relations among the original letters correspond to the relations among side lengths and diagonals of a polygon. By embedding the letters into a (2n+1)‑gon, each mutable cluster variable is identified with a diagonal, while frozen variables correspond to boundary edges. The quiver obtained from the clockwise orientation of the triangles reproduces the mutation rules of the corresponding A‑type cluster algebra. This geometric picture provides a visual realisation of the “cluster adjacency” property that has been crucial in scattering‑amplitude bootstrap programs.

For the one‑loop bubble correlator, the analysis reveals a different structure. The bubble diagram can be decomposed into two independent triangular sub‑graphs, each giving rise to an A₃ cluster algebra. The full alphabet of the bubble correlator is therefore the union of the two A₃ alphabets. This demonstrates that loop corrections can lead to a direct sum of cluster algebras rather than a single higher‑rank algebra.

Overall, the paper establishes that the singularity structure of cosmological correlators in a simple scalar theory is governed by A‑type cluster algebras: tree‑level ladders by A_{2(n−1)} and the one‑loop bubble by two copies of A₃. By mapping the symbol letters to cluster coordinates, the authors provide a powerful algebraic framework that can be used to constrain and bootstrap analytic expressions for cosmological correlators, much as cluster algebras have done for planar N=4 super‑Yang‑Mills amplitudes. The work opens several avenues for future research, including extensions to higher loops, non‑planar topologies, and other field content (e.g., tensors or gravitons), as well as the exploration of non‑simply‑laced cluster algebras (G₂, F₄) that may appear in more intricate cosmological observables.