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The Sklyanin Bracket and Cluster Adjacency at All Multiplicity

John Golden, Andrew J. McLeod, Marcus Spradlin, Anastasia Volovich

TL;DR

This paper develops the Sklyanin Poisson bracket on Gr(4,n) as a practical criterion for cluster adjacency of symbol entries in planar N=4 sYM amplitudes. It shows that, under appropriate regularizations, all one- and two-loop MHV amplitudes satisfy cluster adjacency, and that this property automatically enforces the extended Steinmann relations across arbitrary multiplicities. By connecting adjacency to half-integer Sklyanin brackets, the authors provide both a computationally efficient test and a conceptual link between cluster structures and the analytic structure of amplitudes. The work suggests a robust, regulator-insensitive principle guiding the bootstrap of high-multiplicity amplitudes and highlights open questions about infinite Gr(4,n) cluster algebras and possible generalizations to other amplitude sectors.

Abstract

We argue that the Sklyanin Poisson bracket on Gr(4,n) can be used to efficiently test whether an amplitude in planar ${\cal{N}}=4$ 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.

The Sklyanin Bracket and Cluster Adjacency at All Multiplicity

TL;DR

This paper develops the Sklyanin Poisson bracket on Gr(4,n) as a practical criterion for cluster adjacency of symbol entries in planar N=4 sYM amplitudes. It shows that, under appropriate regularizations, all one- and two-loop MHV amplitudes satisfy cluster adjacency, and that this property automatically enforces the extended Steinmann relations across arbitrary multiplicities. By connecting adjacency to half-integer Sklyanin brackets, the authors provide both a computationally efficient test and a conceptual link between cluster structures and the analytic structure of amplitudes. The work suggests a robust, regulator-insensitive principle guiding the bootstrap of high-multiplicity amplitudes and highlights open questions about infinite Gr(4,n) cluster algebras and possible generalizations to other amplitude sectors.

Abstract

We argue that the Sklyanin Poisson bracket on Gr(4,n) can be used to efficiently test whether an amplitude in planar 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.

Paper Structure

This paper contains 14 sections, 44 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Cluster Algebras and Poisson Brackets
  3. Cluster Coordinates
  4. The Sklyanin Bracket on $\mathcal{X}$-Coordinates
  5. The Sklyanin Bracket on $\mathcal{A}$-Coordinates
  6. MHV Amplitudes
  7. The BDS-Like Normalized MHV Amplitude
  8. The Minimally Normalized MHV Amplitude
  9. Cluster Adjacency of MHV Amplitudes
  10. One Loop
  11. Two Loops
  12. Cluster Adjacency and the Steinmann Relations
  13. Conclusion
  14. The Minimally Normalized One-Loop MHV Amplitude

Figures (1)

  • Figure 1: (a) One of the 14 quivers for the $\mathop{\mathrm{Gr}}\nolimits(4,6)$ cluster algebra, with each vertex labeled by its associated $\mathcal{A}$-coordinate and the frozen coordinates in boxes. (b) The same quiver with frozen vertices omitted and with each mutable vertex labeled by its associated $\mathcal{X}$-coordinate.