Split-Helicity Tree Amplitudes and Flag Cluster Algebras

TL;DR

This work addresses whether tree-level split-helicity gluon amplitudes exhibit cluster adjacency with respect to the partial flag cluster algebra $\mathcal{F\ell}_{2,n-2;n}$. It develops a constructive framework using zigzag diagrams to index BCFW-type terms and assigns a set of good permutations via rooted arc-permutation rules, tested through the Sklyanin bracket after embedding into Gr$(n-2,2n-4)$. Empirical evidence up to $n\le 11$ supports a conjecture that each term is cluster-adjacent precisely under its corresponding good permutations, with exact adjacency for all good permutations in some cases and selective validity in others. The work connects cluster adjacency to on-shell diagrams and amplituhedron tilings, and it discusses potential split-helicity generalizations as well as tensions in non-chiral superspace, outlining clear directions for extending the framework to broader amplitude contexts.

Abstract

Recent work has uncovered a connection between the symbol letters of general massless scattering and (permutations of) cluster variables of partial flag varieties. In this paper we explore the cluster adjacency of tree-level gluon amplitudes, specifically focusing on split-helicity amplitudes which can be written in closed form in terms of zigzag diagrams. We check in several cases, and conjecture in general, that the poles in each term satisfy cluster adjacency under a set of permutations that is built from arc permutations of the corresponding zigzag.

Figures (2)

• Figure 1: From left to right: the initial cluster of 5-point spinor helicity variety, the initial cluster of $\mathcal{F\ell}_{2,3;5}$ and the embedding of $\mathcal{F\ell}_{2,3;5}$ into $\text{Gr}(3,6)$, with frozen vertices boxed. In this example, the cluster algebra of $\mathcal{F\ell}_{2,3;5}$ is isomorphic to that of $\text{Gr}(3,6)$, but this is not the case in general.
• Figure 2: One of the 1716 zigzag diagrams that contribute to the 17-particle N${}^7$MHV amplitude. The initial labeled points are shown in black. The zigzag divides the area between the horizontal lines into 2 boundary and 5 internal regions. There are 10 internal points, $n_R=3$ points in the right boundary region, and $n_L=4$ points in the left. The rule (\ref{['eq:markrule']}) marks boundary points 11 and 15. The target integers $\{\color{red}4\color{black},\ldots,\color{red}13\color{black}\}$ are assigned uniquely to internal points as shown in red. There are $4 \times 8 = 32$ valid assignments of the remaining target integers $\{\color{red}1\color{black},\color{red}2\color{black},\color{red}3\color{black}\}$ and $\{\color{red}14\color{black},\color{red}15\color{black},\color{red}16\color{black},\color{red}17\color{black}\}$ to points in the right and left boundary regions respectively, and so a total of 64 'good permutations' associated to this zigzag diagram.