Generalised Cluster Adjacency for Cosmology

Abstract

In this paper we study the cluster algebraic properties of wavefunction coefficients for massless scalar theories in de Sitter cosmology. We show that the symbol of the wavefunction coefficient of the $n$-site path graph $P_n$ obeys a generalisation of cluster adjacency, where all letters in a given word belong to the same cluster of an $A_{2n-3}$ algebra, with certain additional constraints on the order of the letters. We call this property the ordered single cluster condition, and provide its physical interpretation. This condition is stronger than the usual cluster adjacency obeyed by neighbouring letters, and imposes stronger constraints for the symbol bootstrap. We also show how any tree graph satisfies a cluster-like structure in terms of tubes and tubings on the underlying graph, which allows for a similar bootstrap approach.

Figures (6)

• Figure 1: Extended $2$-site path graph.
• Figure 2: Hexagon corresponding to extended $P_3$.
• Figure 3: General exchange relation for the tube functions on the extended graph.
• Figure 4: Discontinuity of the cosmology wavefunction.
• Figure 5: Discontinuity condition for symbols.