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Cosmology meets cluster algebra

Mattia Capuano, Livia Ferro, Tomasz Lukowski, Alessandro Palazio

TL;DR

The paper reveals a deep link between cosmological wavefunction coefficients in power-law FRW cosmologies and type-$A$ cluster algebras. By analyzing path-graph contributions in de Sitter space, the authors show that the symbol letters are region variables that map to the $A_{2n-2}$ $\mathcal{X}$-coordinates via simplicial coordinates on $\mathcal{M}_{0,2n+1}$, enabling representation of wavefunction coefficients as cluster functions. They develop a general framework using graph tubings to derive differential equations and symbols, then illustrate with explicit computations for $P_2$ and $P_3$, yielding polylogarithmic expressions in terms of Goncharov functions. The results offer a new algebraic perspective on cosmological observables and suggest natural extensions to more complex graphs and the exploration of cluster-adjacency in cosmology.

Abstract

In this paper we explore the mathematical properties of wavefunction coefficients in power-law FRW cosmologies, and establish their relation to cluster algebras. We focus on the particular contributions to the wavefunction coefficient coming from the path Feynman graphs, and show that the singularities of the wavefunction associated with a $n$-site path graph are related to the $\mathcal{X}$-coordinates of the cluster algebra $A_{2n-2}$. To establish this relation, we consider the symbol of the de Sitter wavefunction coefficients and show that the letters appearing there are the region variables associated to tubings on the path graph. These variables can be rewritten as simplicial coordinates of the moduli space $\mathcal{M}_{0,2n+1}$ and therefore identified with the $\mathcal{X}$-coordinates of type-$A_{2n-2}$ cluster algebras. We use this result to compute the wavefunction coefficients in terms of cluster functions.

Cosmology meets cluster algebra

TL;DR

The paper reveals a deep link between cosmological wavefunction coefficients in power-law FRW cosmologies and type- cluster algebras. By analyzing path-graph contributions in de Sitter space, the authors show that the symbol letters are region variables that map to the -coordinates via simplicial coordinates on , enabling representation of wavefunction coefficients as cluster functions. They develop a general framework using graph tubings to derive differential equations and symbols, then illustrate with explicit computations for and , yielding polylogarithmic expressions in terms of Goncharov functions. The results offer a new algebraic perspective on cosmological observables and suggest natural extensions to more complex graphs and the exploration of cluster-adjacency in cosmology.

Abstract

In this paper we explore the mathematical properties of wavefunction coefficients in power-law FRW cosmologies, and establish their relation to cluster algebras. We focus on the particular contributions to the wavefunction coefficient coming from the path Feynman graphs, and show that the singularities of the wavefunction associated with a -site path graph are related to the -coordinates of the cluster algebra . To establish this relation, we consider the symbol of the de Sitter wavefunction coefficients and show that the letters appearing there are the region variables associated to tubings on the path graph. These variables can be rewritten as simplicial coordinates of the moduli space and therefore identified with the -coordinates of type- cluster algebras. We use this result to compute the wavefunction coefficients in terms of cluster functions.

Paper Structure

This paper contains 14 sections, 53 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Note.
  3. Cosmological Background
  4. Symbols for de Sitter
  5. $A_n$ Cluster algebras and $\mathcal{M}_{0,n}$
  6. Cluster algebras for path graphs
  7. Examples
  8. Path graph $\mathbf{P_2}$
  9. Path graph $\mathbf{P_3}$
  10. Applications
  11. Path graph $P_2$
  12. Path graph $P_3$
  13. Conclusions
  14. Acknowledgements

Figures (1)

  • Figure 1: A flip of the diagonal $(14)$ in the quadrilateral $(1345)$ corresponds to a mutation of the $a_2$ variable in the seed cluster $\{a_1,a_2,a_3\}$ into the cluster {$a_1,a_4,a_3$}. The cross-ratios associated to each quadrilateral \ref{['eq: cross ratios']} mutate like $\mathcal{X}$-coordinates.