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Cluster Bootstrap for Cosmological Correlators

Shruti Paranjape, Marcos Skowronek, Marcus Spradlin, Anastasia Volovich, He-Chen Weng

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 $ε$ 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.

Cluster Bootstrap for Cosmological Correlators

Abstract

We show that cosmological wavefunction coefficients associated with -site chain and loop graphs for a cubic scalar theory in de Sitter spacetime have symbol alphabets given by subsets of and 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 expansion. We use this information as bootstrap input to show that de Sitter symbols for are uniquely determined by simple physical constraints.
Paper Structure (14 sections, 35 equations, 3 figures, 1 table)

This paper contains 14 sections, 35 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Review
  3. Symbols for dS wavefunction coefficients
  4. Cluster variables and compatibility
  5. Chain Graphs
  6. Symbol alphabets from A-type cluster algebras
  7. Cluster adjacency
  8. Example: 2-site chain
  9. Example: 3-site chain
  10. Bootstrap
  11. Polygonal Graphs
  12. Symbol alphabets from B-type cluster algebras
  13. Cluster adjacency
  14. Example: 2-site loop

Figures (3)

  • Figure 1: Examples of a valid (left) and forbidden (right) pair of tubings corresponding to adjacent entries in the symbol for the 5-site chain graph.
  • Figure 2: Left: example of a symmetric triangulation of the octagon, corresponding to a cluster in $B_3$. Right: example of an invalid triangulation.
  • Figure 3: Example of a triangulation of the pentagon associated with the $A_2$ cluster algebra of the 2-site chain. Each chord $\Delta_{i,j}$ has been labeled by its associated Plücker variable obtained from plugging (\ref{['eq:2param']}) into (\ref{['eq: A type matrix param']}).