de Sitter Wavefunction from Quadrangular Polylogarithms: Chain Graphs

Abstract: We present an explicit formula for the $n$-site chain graph contribution to the cosmological wavefunction for conformally coupled $φ^3$ 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.

• The paper derives closed-form de Sitter chain graph wavefunctions using quadrangular polylogarithms, demonstrating total cluster compatibility in the analytic structure.
• It leverages explicit analytic techniques in conformally coupled φ³ theory to transform complex integrals into tractable transcendental functions.
• The results bridge advanced mathematical frameworks such as cluster algebras and Hopf algebra structures with practical computations in quantum cosmology.

de Sitter Wavefunction from Quadrangular Polylogarithms and Chain Graphs

Context and Motivation

This paper establishes explicit analytic expressions for the $n$-site chain graph contributions to the cosmological wavefunction in conformally coupled $\phi^3$ theory on de Sitter (dS) space. The authors leverage recent progress uncovering that the symbol of these wavefunction coefficients satisfies total compatibility with respect to $A_{2n-2}$ cluster algebras. Total compatibility sharply constrains the analytic structure: all symbol letters in any term are mutually compatible and reside in a single cluster, a condition much stronger than conventional cluster adjacency, which only requires compatibility between adjacent symbol letters. This mathematical property has allowed a complete basis for such functions, namely Rudenko's quadrangular polylogarithms, to be utilized as the natural language for expressing cosmological wavefunction coefficients. The connection provides a physical instantiation of quadrangular polylogarithms, previously studied in pure mathematics in contexts ranging from Goncharov's depth conjecture to hyperbolic volume formulas.

Review of Two- and Three-Site Chain Wavefunction Coefficients

The paper begins by demonstrating novel quadrangular polylogarithm representations for cosmological chain wavefunctions in low multiplicities. Symbols for the two-site and three-site chains are systematically uplifted to invariant cross-ratios via Plücker coordinates on Grassmannians $G(2,5)$ and $G(2,7)$, revealing a concise geometric structure matching quadrangular polylogarithms, with explicit formulas involving dilogarithms and combinations thereof. This is shown to be manifestly cluster compatible, sometimes requiring the introduction or rearrangement of “spurious” symbol variables—ones absent from the gauge-fixed forms but necessary for compatibility. The integration of these symbols yields expressions with no extraneous “beyond symbol” terms and clear geometric generalizability.

For example, the two-site coefficient $\psi_2$ can be written as a sum/difference of $F_2(a,b,c,d)$ functions—rooted quadrilateral polylogarithms. The three-site chain coefficient $\psi_3$ is expressed as a linear combination of rooted hexagonal polylogarithms $F_3(a,b,c,d,e,f)$, with the structure dictated uniquely by the requirement of total compatibility and matching with the symbolic recursion.

Rudenko's Quadrangular Polylogarithms

Quadrangular polylogarithms $\QLi_m^\pm$ are constructed recursively on alternating $\phi^3$0-gons using combinatorial quadrangulation and the Connes-Kreimer Hopf algebra of rooted trees, assigning weights to symbol letters in correspondence with cross-ratios determined by polygon parity. The defining properties of these functions are:

• Their symbol satisfies total compatibility with respect to the $\phi^3$1 cluster algebra;
• They admit explicit recursive constructions via arborification, associating weighted symbol letters to quadrangulations;
• Their coproducts and functional derivatives are controlled in terms of lower-weight quadrangular polylogarithms and cross-ratios, crucial for recursive solution strategies.

For chain graphs with $\phi^3$2 sites, quadrangular polylogarithms provide exhaustive bases for totally compatible symbols, enabling analytic reduction of highly nested integrals into a tractable set of transcendental functions.

Cosmological Chain Graph Recursion

Wavefunction coefficients for chain graphs obey recursive differential equations, initially formulated in terms of discontinuities or derivatives with respect to external variables. By embedding the kinematic variables into $\phi^3$3 and recasting shifted arguments in the recursion as column deletions in the Grassmannian matrix, the recursion is transformed into a form precisely aligned with the coproduct structure of quadrangular polylogarithms. The final recursion, written in terms of cross-ratios $\phi^3$4, is:

$\phi^3$5

This matches analytic recursion relations for quadrangular polylogarithms and supports the closed-form solution strategy.

Explicit Closed-Form Solution

The central result is an explicit closed-form formula for the $\phi^3$6-site chain graph coefficient:

$\phi^3$7

where $\phi^3$8 is a universal function encoding a sum over all dissections of the $\phi^3$9-gon into even sub-polygons, each term being a product of quadrangular polylogarithms:

$A_{2n-2}$0

The structure is cyclic under translation by two units (reflecting cluster algebra symmetries) but has more subtle behavior under single-site shifts, controlled via functional identities proved inductively. This formula reduces the analytic complexity of chain graph wavefunctions to combinations of well-understood polylogarithmic functions, and holds for all $A_{2n-2}$1, confirmed via inductive proof and soft limit theorems.

Implications and Future Directions

The analytic reduction of cosmological chain graph wavefunctions to quadrangular polylogarithms represents a significant simplification and renders total cluster compatibility in dS wavefunctions manifest. The explicit formulas and recursion harness the full machinery of cluster algebras and the Hopf algebra structure of polylogarithms, bridging concepts previously confined to planar amplitudes in $A_{2n-2}$2 SYM with the physics of curved cosmology.

Practically, this sets the stage for generalizing the analysis to more elaborate cosmological graphs, including loop configurations and those governed by $A_{2n-2}$3-type cluster algebras, as well as for addressing questions regarding the persistence of total compatibility in the presence of mass or higher-order corrections. The methodology may extend to the study of correlators, which are known to be analytically simpler than wavefunction coefficients.

Theoretically, the identification of total cluster compatibility, stronger than cluster adjacency, with physically realized functions suggests a deeper universality in the analytic constraints governing quantum fields on varied backgrounds, not merely a curiosity of flat-space gauge theory amplitudes. The explicit realization of quadrangular polylogarithms in cosmology offers new tools for computations and for probing the mathematical structure of quantum field observables in curved space.

Conclusion

This work provides a rigorous bridge between recent mathematical advances in cluster polylogarithms and practical computations in cosmological wavefunctions. The explicit, recursive, closed-form solution for chain graph coefficients in de Sitter cosmology leverages the strong analytic constraints of total cluster compatibility, realized through quadrangular polylogarithms. The results facilitate a deeper understanding of the analytic structure underpinning quantum field correlators on curved backgrounds, with substantial implications for both future computational strategies and the theoretical formulation of cosmological bootstrap and cluster algebra approaches (2605.06542).