Cluster Algebras and the Subalgebra Constructibility of the Seven-Particle Remainder Function

TL;DR

The article investigates how cluster algebras organize the nonclassical content of the seven-particle two-loop MHV remainder in planar ${\cal N}=4$ sYM. It demonstrates that ${\rm Gr}(4,7)$ contains corank-one subalgebras ${D_5}$ and ${A}_5$ whose associated cluster polylogarithms, reduced further to ${A_4}$ blocks, canonically assemble the nonclassical part via automorphism constraints. A central technical tool is the ${A_2}$-basis function ${f_{A_2}}$ and its extensions to ${A_3}$, ${A_4}$, and beyond, enabling nested decompositions that are unique up to overall scale. The results reveal a remarkably structured, subalgebra–driven decomposition of the seven-point remainder, suggesting a general pattern for higher multiplicities and hinting at deeper combinatorial underpinnings of amplitudes in ${\cal N}=4$ sYM. The methods set the stage for extending subalgebra constructibility to higher points and to NMHV sectors, with potential implications for encoding analytic structure via cluster algebras and their automorphisms.

Abstract

We review various aspects of cluster algebras and the ways in which they appear in the study of loop-level amplitudes in planar ${\cal N} = 4$ supersymmetric Yang-Mills theory. In particular, we highlight the different forms of cluster-algebraic structure that appear in this theory's two-loop MHV amplitudes---considered as functions, symbols, and at the level of their Lie cobracket---and recount how the `nonclassical' part of these amplitudes can be decomposed into specific functions evaluated on the $A_2$ or $A_3$ subalgebras of Gr$(4,n)$. We then extend this line of inquiry by searching for other subalgebras over which these amplitudes can be decomposed. We focus on the case of seven-particle kinematics, where we show that the nonclassical part of the two-loop MHV amplitude is also constructible out of functions evaluated on the $D_5$ and $A_5$ subalgebras of Gr$(4,7)$, and that these decompositions are themselves decomposable in terms of the same $A_4$ function. These nested decompositions take an especially canonical form, which is dictated in each case by constraints arising from the automorphism group of the parent algebra.

Figures (7)

• Figure 1: All possible triangulations of the pentagon. Mutating on the red chord moves you clockwise around the figure.
• Figure 2: A triangulation of the hexagon along with its assocated $\mathcal{A}$-coordinate and $\mathcal{X}$-coordinate seed quivers.
• Figure 3: The cluster polytope of $A_3\simeq\mathop{\mathrm{Gr}}\nolimits(2,6)$, in which each cluster is represented by a triangulation of the hexagon.
• Figure 4: The cluster polytope of $\mathop{\mathrm{Gr}}\nolimits(4,7) \simeq E_6$.
• Figure 5: The $\mathcal{A}$-coordinate seed quiver (a) and $\mathcal{X}$-coordinate seed quiver (b) for $\mathop{\mathrm{Gr}}\nolimits(4,6)$.