Symbol Alphabets from Tensor Diagrams

TL;DR

This paper introduces a novel approach using tensor diagrams and polytope geometry to analyze symbol alphabets in planar N=4 Yang-Mills theory, connecting algebraic and rational letters to Grassmannian structures.

Contribution

It establishes a new framework linking tensor diagrams, cluster variables, and algebraic symbol letters via polytopes associated with Grassmannians, extending previous analyses to nine particles.

Findings

Reproduces known results for n ≤ 8 particles.

Identifies all rational and algebraic letters for nine-particle amplitudes.

Shows the polytope C†(4,9) encodes all relevant symbol letters.

Abstract

We propose to use tensor diagrams and the Fomin-Pylyavskyy conjectures to explore the connection between symbol alphabets of $n$-particle amplitudes in planar $\mathcal{N}=4$ Yang-Mills theory and certain polytopes associated to the Grassmannian G(4, $n$). We show how to assign a web (a planar tensor diagram) to each facet of these polytopes. Webs with no inner loops are associated to cluster variables (rational symbol letters). For webs with a single inner loop we propose and explicitly evaluate an associated web series that contains information about algebraic symbol letters. In this manner we reproduce the results of previous analyses of $n \le 8$, and find that the polytope $\mathcal{C}^\dagger(4,9)$ encodes all rational letters, and all square roots of the algebraic letters, of known nine-particle amplitudes.