Towards the Amplituhedron Volume

Abstract

It has been recently conjectured that scattering amplitudes in planar N=4 super Yang-Mills are given by the volume of the (dual) amplituhedron. In this paper we show some interesting connections between the tree-level amplituhedron and a special class of differential equations. In particular we demonstrate how the amplituhedron volume for NMHV amplitudes is determined by these differential equations. The new formulation allows for a straightforward geometric description, without any reference to triangulations. Finally we discuss possible implications for volumes related to generic N^kMHV amplitudes.

21 pages; v2: version published in JHEP

References in corpus (4)

• Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory
• The Yangian origin of the Grassmannian integral
• Harmonic R-matrices for Scattering Amplitudes and Spectral Regularization
• N=4 Scattering Amplitudes and the Deformed Grassmannian

Cited by in corpus (32)

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• Multi-Loop Positivity of the Planar $\mathcal{N}=4$ SYM Six-Point Amplitude
• Nonperturbative Negative Geometries: Amplitudes at Strong Coupling and the Amplituhedron
• Notes on Scattering Amplitudes as Differential Forms
• Positive geometry, local triangulations, and the dual of the Amplituhedron
• Amplituhedron meets Jeffrey-Kirwan Residue
• Grassmannians and form factors with $q^2=0$ in N=4 SYM theory
• Wilson lines, Grassmannians and Gauge Invariant Off-shell Amplitudes in N=4 SYM
• Sign Flip Triangulations of the Amplituhedron
• Amplituhedra, and Beyond
• Towards the Gravituhedron: New Expressions for NMHV Gravity Amplitudes
• On-shell diagrams and the geometry of planar N < 4 SYM theories
• Four dimensional ambitwistor strings and form factors of local and Wilson line operators
• Yangian Symmetry for the Tree Amplituhedron
• Non-planar BCFW Grassmannian Geometries
• Tree-level scattering amplitudes from the amplituhedron
• Triangulations and Canonical Forms of Amplituhedra: a fiber-based approach beyond polytopes
• Poles At Infinity in On-shell Diagrams
• Grassmannian Integral for General Gauge Invariant Off-shell Amplitudes in N=4 SYM
• Triangulation of 2-loop MHV Amplituhedron from Sign Flips
• A note on generalized hypergeometric functions, KZ solutions, and gluon amplitudes
• The twistor Wilson loop and the amplituhedron
• Positivity, Grassmannian Geometry and Simplex-like Structures of Scattering Amplitudes
• Positive Geometry for Stringy Scalar Amplitudes
• Logarithms and Volumes of Polytopes
• Bipartite Graphs, On-Shell Diagrams, and Bipartite Field Theories: a Computational Package in Mathematica