Motivic Amplitudes and Cluster Coordinates

TL;DR

The paper shows that motivic amplitudes in planar N=4 SYM are governed by cluster X-coordinates on the kinematic space Conf_n(P^3). By computing the two-loop MHV coproduct for n=7, it demonstrates that all weight-four motivic content can be expressed using X-coordinates, and reveals a deep link to generalized associahedra (E6 for n=7) via the Λ^2 B_2 and B_3 ⊗ F^* components. A novel 40-term trilogarithm functional equation is proven whose arguments are cluster X-coordinates, highlighting a cluster-theoretic origin of functional relations. The work suggests a universal mechanism: two-loop MHV amplitudes' motivic content is encoded by a finite, highly structured subset of X-coordinates tied to the positive Grassmannian and its associahedral geometry, with implications for higher-loop and non-MHV amplitudes.

Abstract

In this paper we study motivic amplitudes--objects which contain all of the essential mathematical content of scattering amplitudes in planar SYM theory in a completely canonical way, free from the ambiguities inherent in any attempt to choose particular functional representatives. We find that the cluster structure on the kinematic configuration space Conf_n(P^3) underlies the structure of motivic amplitudes. Specifically, we compute explicitly the coproduct of the two-loop seven-particle MHV motivic amplitude A_{7,2} and find that like the previously known six-particle amplitude, it depends only on certain preferred coordinates known in the mathematics literature as cluster X-coordinates on Conf_n(P^3). We also find intriguing relations between motivic amplitudes and the geometry of generalized associahedrons, to which cluster coordinates have a natural combinatoric connection. For example, the obstruction to A_{7,2} being expressible in terms of classical polylogarithms is most naturally represented by certain quadrilateral faces of the appropriate associahedron. We also find and prove the first known functional equation for the trilogarithm in which all 40 arguments are cluster X-coordinates of a single algebra. In this respect it is similar to Abel's 5-term dilogarithm identity.

Figures (12)

• Figure 1: The initial quiver for the $\mathop{\mathrm{Gr}}\nolimits(k,n)$ cluster algebra (see refs. 1057.530641215.16012).
• Figure 2: The initial quiver (a) for $\mathop{\mathrm{Gr}}\nolimits(4,6)$ and its parity conjugate (b).
• Figure 3: The initial quiver (a) for $\mathop{\mathrm{Gr}}\nolimits(4,7)$ and its partner (b) which makes the parity conjugation property manifest.
• Figure 4: A triangulation $T$ of the hexagon. One of the edges of the triangulation is marked by $E$.
• Figure 5: A sequence of mutations for five points. At each step the side colored in red gets flipped.

Theorems & Definitions (9)

• Conjecture
• Lemma A.1
• proof
• Proposition A.2
• Lemma A.3
• proof
• proof : Proof of the Proposition.
• Theorem B.1
• proof : Proof of Theorem \ref{['fe3l']}