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 N $$ \mathcal{N} $$ = 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.