The Sklyanin Bracket and Cluster Adjacency at All Multiplicity

Abstract

We argue that the Sklyanin Poisson bracket on Gr(4,n) can be used to efficiently test whether an amplitude in planar $\mathcal{N}=4$ supersymmetric Yang-Mills theory satisfies cluster adjacency. We use this test to show that cluster adjacency is satisfied by all one- and two-loop MHV amplitudes in this theory, once suitably regulated. Using this technique we also demonstrate that cluster adjacency implies the extended Steinmann relations at all particle multiplicities.

arXiv comment25 pages; v2: added reference

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Cited by in corpus (26)

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• 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
• How tropical are seven- and eight-particle amplitudes?
• Bootstrapping a Stress-Tensor Form Factor through Eight Loops
• Lifting Heptagon Symbols to Functions
• Two-loop Octagons, Algebraic Letters and $\bar{Q}$ Equations
• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• Antipodal Self-Duality for a Four-Particle Form Factor
• All-Mass $n$-gon Integrals in $n$ Dimensions
• Cluster Adjacency for m=2 Yangian Invariants
• Yangian Invariants and Cluster Adjacency in N=4 Yang-Mills
• Constraints on Sequential Discontinuities from the Geometry of On-shell Spaces
• A Note on Letters of Yangian Invariants
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• The Two-Loop Remainder Function for Eight and Nine Particles
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• A Finite $S$-Matrix
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• A Novel Algorithm for Nested Summation and Hypergeometric Expansions
• A Note on One-loop Cluster Adjacency in N = 4 SYM
• Weak Separation, Positivity and Extremal Yangian Invariants