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

Abstract

We review various aspects of cluster algebras and the ways in which they appear in the study of loop-level amplitudes in planar $\mathcal{N}=4$ 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.

arXiv comment68 pages, 7 figures, 6 tables

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• Six-Gluon Amplitudes in Planar $\mathcal{N}=4$ Super-Yang-Mills Theory at Six and Seven Loops
• The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N}=4$ SYM Amplitudes
• A Three-Point Form Factor Through Five Loops
• An $S$-Matrix for Massless Particles
• Sequential Discontinuities of Feynman Integrals and the Monodromy Group
• Bootstrapping a Stress-Tensor Form Factor through Eight Loops
• Lifting Heptagon Symbols to Functions
• Two-loop Octagons, Algebraic Letters and $\bar{Q}$ Equations
• Antipodal Self-Duality for a Four-Particle Form Factor
• Three-point energy correlator in $\mathcal{N}=4$ super Yang-Mills Theory
• The Sklyanin Bracket and Cluster Adjacency at All Multiplicity
• All-Mass $n$-gon Integrals in $n$ Dimensions
• Constraints on Sequential Discontinuities from the Geometry of On-shell Spaces
• Yangian Invariants and Cluster Adjacency in N=4 Yang-Mills
• Bootstrapping elliptic Feynman integrals using Schubert analysis
• The Two-Loop Remainder Function for Eight and Nine Particles
• Rooting Out Letters: Octagonal Symbol Alphabets and Algebraic Number Theory
• The Three-loop MHV Octagon from $\bar{Q}$ equations
• Comments on all-loop constraints for scattering amplitudes and Feynman integrals
• The SAGEX Review on Scattering Amplitudes, Chapter 5: Analytic Bootstraps for Scattering Amplitudes and Beyond
• All two-loop MHV remainder functions in multi-Regge kinematics
• One-loop integrand from generalised scattering equations
• A Novel Algorithm for Nested Summation and Hypergeometric Expansions
• Antipodal Symmetry of Two-Loop MHV Amplitudes
• A Note on One-loop Cluster Adjacency in N = 4 SYM