Cluster adjacency beyond MHV

TL;DR

This work extends the notion of cluster adjacency from MHV to non-MHV amplitudes in planar $\mathcal{N}=4$ SYM, linking pole structures in BCFW terms with symbol-level alphabet constraints and the $\bar{Q}$ equation. It develops a unified framework where cluster algebras govern both pole placements and the allowed symbol entries, connecting abelian pole adjacency in tree-level BCFW expansions to non-abelian symbol adjacency via cluster polytopes (e.g., $A_3$ for hexagons and $E_6$ for heptagons). The paper shows that NMHV loop amplitudes exhibit a tight interplay between derivatives of loop functions and their pole structure, with the $\bar{Q}$ constraints reproducing and extending cluster-adjacency relations and Steinmann-type relations. The results for hexagon and heptagon cases motivate generalizations to higher multiplicities and loop orders, and raise questions about extending adjacency notions beyond polylogarithms to elliptic or more general functions, as well as potential applicability beyond planar $\mathcal{N}=4$ theories.

Abstract

We explore further the notion of cluster adjacency, focussing on non-MHV amplitudes. We extend the notion of adjacency to the BCFW decomposition of tree-level amplitudes. Adjacency controls the appearance of poles, both physical and spurious, in individual BCFW terms. We then discuss how this notion of adjacency is connected to the adjacency already observed at the level of symbols of scattering amplitudes which controls the appearance of branch cut singularities. Poles and symbols become intertwined by cluster adjacency and we discuss the relation of this property to the $\bar{Q}$-equation which imposes constraints on the derivatives of the transcendental functions appearing in loop amplitudes.

Figures (25)

• Figure 1: The initial cluster of the Grassmannian series ${\rm Gr}(4,n)$.
• Figure 2: The quiver diagram for the initial cluster for the algebra associated to ${\rm Conf}_6(\mathbb{P}^3)$.
• Figure 3: The $A_3$ Stasheff polytope with six pentagonal faces and three square faces, each labelled with the corresponding $\mathcal{A}$-coordinate. The initial cluster corresponds to the vertex at the top left corner at the intersection of the faces labelled by $\langle 1235 \rangle$, $\langle 1245 \rangle$, $\langle 1345\rangle$. The three-step path leads from the initial cluster to one obtained by a cyclic rotation by one unit.
• Figure 4: The two-brackets $(ij)$ can be identified with chords on a hexagon between the vertices $i$ and $j$. A triangulation of the hexagon then corresponds to a cluster of the $A_3$ or ${\rm Conf}_6(\mathbb{P}^3)$ polytope. Above is shown the triangulation corresponding to the initial cluster of Fig. \ref{['hexinitial']} comprised of the chords $(26) = \langle 1345 \rangle$, $(36) = \langle 1245 \rangle$ and $(46) = \langle 1235 \rangle$ together with the six edges which correspond to the frozen nodes.
• Figure 5: The Stasheff polytope for ${\rm Conf}_6(\mathbb{P}^3) \cong \mathcal{M}_{0,6}$ with the clusters labelled by the different triangulations of a hexagon.