Hedgehog Bases for A_n Cluster Polylogarithms and An Application to Six-Point Amplitudes

TL;DR

The paper develops Hedgehog bases: weight-$k$ Goncharov polylogarithms whose symbol alphabets are $A_n$ cluster coordinates, enabling cluster-structure-aware representations of scattering amplitudes. Building on Radford's Lyndon-word framework and Brown’s moduli-space insights, the authors prove that hedgehogs—subsets associated with $A_{n-1}$ subalgebras—provide bases for the space of $A_n$ cluster functions. They explicitly construct the Hedgehog Basis for $A_3$, enumerate the six hedgehogs (and six anti-hedgehogs), and apply this to recast the 2-loop 6-particle NMHV amplitude, obtaining a shorter, cluster-structure-prominent expression. A key result is the full weight-4 amplitude expressed in a 416-term hedgehog basis, obtained by matching the symbol and fixing beyond-the-symbol terms numerically. The work highlights both the practical benefits for amplitude simplification and the theoretical alignment between cluster algebras and polylogarithmic function bases, while outlining limitations and avenues for extending the approach to other algebras and analytic constraints.

Abstract

Multi-loop scattering amplitudes in N=4 Yang-Mills theory possess cluster algebra structure. In order to develop a computational framework which exploits this connection, we show how to construct bases of Goncharov polylogarithm functions, at any weight, whose symbol alphabet consists of cluster coordinates on the $A_n$ cluster algebra. Using such a basis we present a new expression for the 2-loop 6-particle NMHV amplitude which makes some of its cluster structure manifest.

Figures (5)

• Figure 2: (A) The five seeds for the $A_2$ algebra, with each node labeled by its associated cluster variable. (B) The exchange graph, showing how to move from one seed to another by mutation.
• Figure 3: The factorization graph for $A_2$. Each vertex is one of the 10 cluster coordinates on the $A_2$ cluster algebra (see eq. (\ref{['eq:A2coordinates']})), and two vertices $x_i, x_j$ are connected by an edge if $x_i{-}x_j$ factors into a product of cluster coordinates. Each of the 10 pairs of connected vertices, for example $\{1/x_2, x_5\}$, is a 2-clique.
• Figure 4: The factorization graph for $A_3$. Each vertex represents one of the 30 cluster coordinates on the $A_3$ cluster algebra, although to avoid clutter only 15 of the coordinates are labeled; the other 15 are reciprocals of the ones shown. Two vertices $x_i, x_j$ are connected by an edge if $x_i{-}x_j \in M_{A_3}$. The six circles each pass through 10 vertices and indicate an $A_2$ subalgebra, as shown in figure \ref{['fig:A2factorization']}. There are 12 subgraphs with the topology of a triangle, 6 around the outer edge and 6 around the inner edge; these are the 12 3-cliques.
• Figure 5: The $A_3$ algebra has six distinct hedgehogs (and six anti-hedgehogs). This figure shows the exchange graph for $A_3$, with one of its six pentagonal $A_2$ subalgebras highlighted. The "spines" of this $\mathcal{X}(A_3,A_2)$ hedgehog are the red edges connecting this $A_2$ to the rest of $A_3$. Specifically, this $\mathcal{X}(A_3,A_2)$ is the set of 3 $\mathcal{X}_{A_3}$ cluster coordinates associated to these 5 outward directed red edges.
• Figure 6: (A) The exchange graph for $A_2$, with the two vertices on the bottom row constituting an $A_1$ subalgebra. The hedgehog $\mathcal{X}(A_2,A_1)$ contains the two cluster variables $1/x_{i-1}$, $x_{i+1}$ associated to the edges emanating away from the subalgebra. (B) The same exchange graph, but with each vertex showing the associated pentagon triangulation.