The Geometry of On-Shell Diagrams

Abstract

The fundamental role of on-shell diagrams in quantum field theory has been recently recognized. On-shell diagrams, or equivalently bipartite graphs, provide a natural bridge connecting gauge theory to powerful mathematical structures such as the Grassmannian. We perform a detailed investigation of the combinatorial and geometric objects associated to these graphs. We mainly focus on their relation to polytopes and toric geometry, the Grassmannian and its stratification. Our work extends the current understanding of these connections along several important fronts, most notably eliminating restrictions imposed by planarity, positivity, reducibility and edge removability. We illustrate our ideas with several explicit examples and introduce concrete methods that considerably simplify computations. We consider it highly likely that the structures unveiled in this article will arise in the on-shell study of scattering amplitudes beyond the planar limit. Our results can be conversely regarded as an expansion in the understanding of the Grassmannian in terms of bipartite graphs.

85 pages, 29 figures

References in corpus (7)

• N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals
• The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM
• The Master Space of N=1 Gauge Theories
• ABJM amplitudes and the positive orthogonal grassmannian
• The Grassmannian and the Twistor String: Connecting All Trees in N=4 SYM
• Bipartite Field Theories from D-Branes
• Scattering Amplitudes and Toric Geometry

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• Full Colour for Loop Amplitudes in Yang-Mills Theory
• Gauge Theories and Dessins d'Enfants: Beyond the Torus
• On-shell diagrams and the geometry of planar N < 4 SYM theories
• Twin Theories, Polytope Mutations and Quivers for GTPs
• D-brane Instantons as Gauge Instantons in Orientifolds of Chiral Quiver Theories
• Non-planar BCFW Grassmannian Geometries
• Bipartite field theories and D-brane instantons
• Poles At Infinity in On-shell Diagrams
• BFT${}_2$: a General Class of $2d$ $\mathcal{N}=(0,2)$ Theories, 3-Manifolds and Toric Geometry
• Permutation Relations of Yangian Invariants, Unitarity Cuts, and Scattering Amplitudes
• Top-forms of Leading Singularities in Nonplanar Multi-loop Amplitudes
• Positivity, Grassmannian Geometry and Simplex-like Structures of Scattering Amplitudes
• Boundary Measurement Matrices for Directed Networks on Surfaces
• Consistent Dimer Models on Surfaces with Boundary