Anatomy of the Amplituhedron

Abstract

We initiate a comprehensive investigation of the geometry of the amplituhedron, a recently found geometric object whose volume calculates the integrand of scattering amplitudes in planar N=4 SYM theory. We do so by introducing and studying its stratification, focusing on four-point amplitudes. The new stratification exhibits interesting combinatorial properties and positivity is neatly captured by permutations. As explicit examples, we find all boundaries for the two and three loop amplitudes and related geometries. We recover the stratifications of some of these geometries from the singularities of the corresponding integrands, providing a non-trivial test of the amplituhedron/scattering amplitude correspondence. We finally introduce a deformation of the stratification with remarkably simple topological properties.

arXiv comment74 pages, 15 figures

References in corpus (32)

• Direct Proof Of Tree-Level Recursion Relation In Yang-Mills Theory
• Gluon scattering amplitudes at strong coupling
• Generalized Unitarity and One-Loop Amplitudes in N=4 Super-Yang-Mills
• Iteration of Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory at Three Loops and Beyond
• Transcendentality and Crossing
• The N=4 SYM Integrable Super Spin Chain
• MHV Vertices And Tree Amplitudes In Gauge Theory
• The Amplituhedron
• Total positivity, Grassmannians, and networks
• Magic identities for conformal four-point integrals
• Local Integrals for Planar Scattering Amplitudes
• Dual Superconformal Invariance, Momentum Twistors and Grassmannians
• The Complete Planar S-matrix of N=4 SYM as a Wilson Loop in Twistor Space
• Maximally Supersymmetric Planar Yang-Mills Amplitudes at Five Loops
• Motivic Amplitudes and Cluster Coordinates
• Analytic result for the two-loop six-point NMHV amplitude in N=4 super Yang-Mills theory
• The Grassmannian Origin Of Dual Superconformal Invariance
• Into the Amplituhedron
• Bipartite Field Theories: from D-Brane Probes to Scattering Amplitudes
• The Soft-Collinear Bootstrap: N=4 Yang-Mills Amplitudes at Six and Seven Loops
• ABJM amplitudes and the positive orthogonal grassmannian
• Harmonic R-matrices for Scattering Amplitudes and Spectral Regularization
• Unraveling L_{n,k}: Grassmannian Kinematics
• Matching polytopes, toric geometry, and the non-negative part of the Grassmannian
• N=4 Scattering Amplitudes and the Deformed Grassmannian
• On Yangian-invariant regularisation of deformed on-shell diagrams in N=4 super-Yang-Mills theory
• Bootstrapping six-gluon scattering in planar $\mathcal{N}=4$ super-Yang-Mills theory
• Scattering Amplitudes and Toric Geometry
• Cluster Polylogarithms for Scattering Amplitudes
• Cluster Algebras and the Positive Grassmannian
• Cluster Transformations from Bipartite Field Theories
• Double-dimers, the Ising model and the hexahedron recurrence

Cited by in corpus (6)

• Positive Amplitudes In The Amplituhedron
• Evidence for a Nonplanar Amplituhedron
• On-shell diagrammatics and the perturbative structure of planar gauge theories
• Non-Planar On-Shell Diagrams
• Logarithmic Singularities and Maximally Supersymmetric Amplitudes
• Orientations of BCFW Charts on the Grassmannian