# The Sklyanin Bracket and Cluster Adjacency at All Multiplicity

## TL;DR

This paper demonstrates that the Sklyanin Poisson bracket can efficiently verify cluster adjacency in planar N=4 SYM amplitudes, confirming its validity at low loops and linking it to Steinmann relations across all multiplicities.

## Contribution

It introduces a novel application of the Sklyanin Poisson bracket to test cluster adjacency in scattering amplitudes, establishing its validity at one- and two-loop levels and its connection to Steinmann relations.

## Key findings

- Cluster adjacency holds for all one- and two-loop MHV amplitudes.
- The Sklyanin Poisson bracket effectively tests cluster adjacency.
- Cluster adjacency implies the extended Steinmann relations at all multiplicities.

## 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.