Symbol Alphabets from Tensor Diagrams

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 ${\mathcal{C}}^{†}(4,9)$ encodes all rational letters, and all square roots of the algebraic letters, of known nine-particle amplitudes.

44 pages; v2: quite a few individually minor corrections and improvements

References in corpus (7)

• Classical Polylogarithms for Amplitudes and Wilson Loops
• Total positivity, Grassmannians, and networks
• Cluster algebras for Feynman integrals
• Scattering Equations: From Projective Spaces to Tropical Grassmannians
• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• Notes on cluster algebras and some all-loop Feynman integrals
• Algebraic branch points at all loop orders from positive kinematics and wall crossing

Cited by in corpus (13)

• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• The elliptic double box and symbology beyond polylogarithms
• Antipodal Self-Duality for a Four-Particle Form Factor
• Truncated cluster algebras and Feynman integrals with algebraic letters
• Symbology for elliptic multiple polylogarithms and the symbol prime
• Constraints on Sequential Discontinuities from the Geometry of On-shell Spaces
• Schubert Problems, Positivity and Symbol Letters
• Bootstrapping elliptic Feynman integrals using Schubert analysis
• Towards Analytic Structure of Feynman Parameter Integrals with Rational Curves
• Comments on all-loop constraints for scattering amplitudes and Feynman integrals
• The SAGEX Review on Scattering Amplitudes, Chapter 5: Analytic Bootstraps for Scattering Amplitudes and Beyond
• Non-planar BCFW Grassmannian Geometries
• Notes on Worldsheet-Like Variables for Cluster Configuration Spaces