Non-Planar On-Shell Diagrams

Abstract

We initiate a systematic study of non-planar on-shell diagrams in N=4 SYM and develop powerful technology for doing so. We introduce canonical variables generalizing face variables, which make the dlog form of the on-shell form explicit. We make significant progress towards a general classification of arbitrary on-shell diagrams by means of two classes of combinatorial objects: generalized matching and matroid polytopes. We propose a boundary measurement that connects general on-shell diagrams to the Grassmannian. Our proposal exhibits two important and non-trivial properties: positivity in the planar case and it matches the combinatorial description of the diagrams in terms of generalized matroid polytopes. Interestingly, non-planar diagrams exhibit novel phenomena, such as the emergence of constraints on Plucker coordinates beyond Plucker relations when deleting edges, which are neatly captured by the generalized matching and matroid polytopes. This behavior is tied to the existence of a new type of poles in the on-shell form at which combinations of Plucker coordinates vanish. Finally, we introduce a prescription, applicable beyond the MHV case, for writing the on-shell form as a function of minors directly from the graph.

85 pages, 34 figures. v3: section 7 improved, typos corrected, references updated

References in corpus (21)

• Gluon scattering amplitudes at strong coupling
• Classical Polylogarithms for Amplitudes and Wilson Loops
• Conformal properties of four-gluon planar amplitudes and Wilson loops
• MHV Amplitudes in N=4 Super Yang-Mills and Wilson Loops
• Total positivity, Grassmannians, and networks
• Gauge Amplitude Identities by On-shell Recursion Relation in S-matrix Program
• Bootstrapping an NMHV amplitude through three loops
• A Symbol of Uniqueness: The Cluster Bootstrap for the 3-Loop MHV Heptagon
• The Complete Four-Loop Four-Point Amplitude in N=4 Super-Yang-Mills Theory
• Positive Amplitudes In The Amplituhedron
• On the Singularity Structure of Maximally Supersymmetric Scattering Amplitudes
• Harmonic R-matrices for Scattering Amplitudes and Spectral Regularization
• A Cluster Bootstrap for Two-Loop MHV Amplitudes
• From Twistor String Theory To Recursion Relations
• The Grassmannian and the Twistor String: Connecting All Trees in N=4 SYM
• Three Applications of a Bonus Relation for Gravity Amplitudes
• Matching polytopes, toric geometry, and the non-negative part of the Grassmannian
• N=4 Scattering Amplitudes and the Deformed Grassmannian
• Deformed one-loop amplitudes in N = 4 super-Yang-Mills theory
• Anatomy of the Amplituhedron
• Amplituhedron cells and Stanley symmetric functions

Cited by in corpus (8)

• Charting Class ${\mathcal{S}}_k$ Territory
• Dual Conformal Structure Beyond the Planar Limit
• On Soft Theorems And Form Factors In N=4 SYM Theory
• On-shell diagrammatics and the perturbative structure of planar gauge theories
• Top-forms of Leading Singularities in Nonplanar Multi-loop Amplitudes
• A note on NMHV form factors from the Graßmannian and the twistor string
• Boundary Measurement Matrices for Directed Networks on Surfaces
• Simplex-like Structures of Maximally Supersymmetric Scattering Amplitudes