Unwinding the Amplituhedron in Binary

Abstract

We present new, fundamentally combinatorial and topological characterizations of the amplituhedron. Upon projecting external data through the amplituhedron, the resulting configuration of points has a specified (and maximal) generalized 'winding number'. Equivalently, the amplituhedron can be fully described in binary: canonical projections of the geometry down to one dimension have a specified (and maximal) number of 'sign flips' of the projected data. The locality and unitarity of scattering amplitudes are easily derived as elementary consequences of this binary code. Minimal winding defines a natural 'dual' of the amplituhedron. This picture gives us an avatar of the amplituhedron purely in the configuration space of points in vector space (momentum-twistor space in the physics), a new interpretation of the canonical amplituhedron form, and a direct bosonic understanding of the scattering super-amplitude in planar N = 4 SYM as a differential form on the space of physical kinematical data.

42 pages, 13 figures

References in corpus (8)

• N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals
• Classical Polylogarithms for Amplitudes and Wilson Loops
• Positive Geometries and Canonical Forms
• Landau Singularities from the Amplituhedron
• The Amplituhedron from Momentum Twistor Diagrams
• The Correlahedron
• Multi-Loop Positivity of the Planar $\mathcal{N}=4$ SYM Six-Point Amplitude
• Tree-level scattering amplitudes from the amplituhedron

Cited by in corpus (71)

• Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet
• The Momentum Amplituhedron
• Cutting Deep Into The Amplituhedron
• On the Positive Geometry of Conformal Field Theory
• Nonperturbative Negative Geometries: Amplitudes at Strong Coupling and the Amplituhedron
• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• Notes on Scattering Amplitudes as Differential Forms
• Boundaries of Amplituhedra and NMHV Symbol Alphabets at Two Loops
• Accordiohedra as positive geometries for generic scalar field theories
• The SAGEX Review on Scattering Amplitudes, Chapter 8: Half BPS correlators
• Symbol Alphabets from Plabic Graphs
• The symbol and alphabet of two-loop NMHV amplitudes from $\bar{Q}$ equations
• Scalar-Scaffolded Gluons and the Combinatorial Origins of Yang-Mills Theory
• On Positive Geometries of Quartic Interactions II : Stokes polytopes, Lower Forms on Associahedra and Worldsheet Forms
• All-Helicity Symbol Alphabets from Unwound Amplituhedra
• Positive geometry, local triangulations, and the dual of the Amplituhedron
• Parity duality for the amplituhedron
• UV cancelations in gravity loop integrands
• The positive geometry for ${\varphi }^p$ interactions
• The orthogonal momentum amplituhedron and ABJM amplitudes
• The momentum amplituhedron of SYM and ABJM from twistor-string maps
• Algebraic branch points at all loop orders from positive kinematics and wall crossing
• Scattering Forms, Worldsheet Forms and Amplitudes from Subspaces
• A Note on Letters of Yangian Invariants
• The SAGEX Review on Scattering Amplitudes, Chapter 7: Positive Geometry of Scattering Amplitudes
• All-Loop Four-Point Aharony-Bergman-Jafferis-Maldacena Amplitudes from Dimensional Reduction of the Amplituhedron
• An Etude on Recursion Relations and Triangulations
• Amplituhedron meets Jeffrey-Kirwan Residue
• Two-Loop Five-Point Two-Mass Planar Integrals and Double Lagrangian Insertions in a Wilson Loop
• Positive Geometries for all Scalar Theories from Twisted Intersection Theory
• Amplituhedra, and Beyond
• Sign Flip Triangulations of the Amplituhedron
• Towards the Gravituhedron: New Expressions for NMHV Gravity Amplitudes
• Notes on polytopes, amplitudes and boundary configurations for Grassmannian string integrals
• Integrated negative geometries in ABJM
• Multi-spin soft bootstrap and scalar-vector Galileon
• Amplituhedron-like geometries
• The Loop Momentum Amplituhedron
• All-loop cuts from the Amplituhedron
• Weights and recursion relations for ${\varphi }^p$ tree amplitudes from the positive geometry
• Gravity loop integrands from the ultraviolet
• Wilson Loop Duality and OPE for Super Form Factors of Half-BPS Operators
• Emergent unitarity, all-loop cuts and integrations from the ABJM amplituhedron
• Towards Analytic Structure of Feynman Parameter Integrals with Rational Curves
• On Positive Geometry and Scattering Forms for Matter Particles
• Multi-Particle Amplitudes from the Four-Point Correlator in Planar N=4 SYM
• Non-planar BCFW Grassmannian Geometries
• The all-loop conjecture for integrands of reggeon amplitudes in N=4 SYM
• $\mathcal{N}=7$ On-Shell Diagrams and Supergravity Amplitudes in Momentum Twistor Space
• Open associahedra and scattering forms
• $\Delta$-Algebra and Scattering Amplitudes
• Bootstrapping octagons in reduced kinematics from $A_2$ cluster algebras
• Triangulation of 2-loop MHV Amplituhedron from Sign Flips
• Emergent unitarity from the amplituhedron
• Poles At Infinity in On-shell Diagrams
• On the Positive Geometry of Quartic Interactions III : One Loop Integrands from Polytopes
• Triangulation-free Trivialization of 2-loop MHV Amplituhedron
• The twistor Wilson loop and the amplituhedron
• Gradient flows, adjoint orbits, and the topology of totally nonnegative flag varieties
• The Amplituhedron BCFW Triangulation
• Connections between Minkowski and Cosmological Correlation Functions
• BCFW tilings and cluster adjacency for the amplituhedron
• Leading singularities and chambers of Correlahedron
• Positive Geometry for Stringy Scalar Amplitudes
• A statistical model for quantum spin and photon number states
• Scattering amplitudes as multi-particle higher-spin charges in the correspondence space
• Quantum deformation of planar amplitudes
• Probing $\Delta L=2$ lepton number violating SMEFT operators at the same-sign muon collider
• The Geometry of BCFW for ABJM Loop Integrands
• Extended DBI and its generalizations from graded soft theorems
• Cyclicity of All Anti-NMHV and N${}^2$MHV Tree Amplitudes in N=4 SYM