de Sitter Wavefunction from Quadrangular Polylogarithms: Chain Graphs

TL;DR

This paper derives an explicit formula for the chain graph contribution to the de Sitter wavefunction in conformally coupled $$ theory, utilizing quadrangular polylogarithms and cluster algebra properties.

Contribution

It introduces a novel explicit formula connecting de Sitter wavefunctions with quadrangular polylogarithms and cluster algebra structures.

Findings

The symbol of the wavefunction satisfies total compatibility with the $A_{2n-2}$ cluster algebra.

Quadrangular polylogarithms form a complete basis for these functions.

The recursive differential equations relate directly to coproduct formulas for quadrangular polylogarithms.

Abstract

We present an explicit formula for the $n$-site chain graph contribution to the cosmological wavefunction for conformally coupled $\phi^3$ theory in de Sitter space. Our result relies on the recent finding that the symbol of this function satisfies total compatibility with respect to the $A_{2n-2}$ cluster algebra, and that Rudenko's quadrangular polylogarithms provide, by construction, a complete basis for such functions. We prove our formula by directly relating a recursive set of differential equations satisfied by these wavefunction coefficients to a recursive coproduct formula for quadrangular polylogarithms.