de Sitter Wavefunction from Quadrangular Polylogarithms: Chain Graphs

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.