Amplituhedra, and Beyond

Abstract

This review is a primer on recently established geometric methods for observables in quantum field theories. The main emphasis is on amplituhedra, i.e. geometries encoding scattering amplitudes for a variety of theories. These pertain to a broader family of geometries called positive geometries, whose basics we review. We also describe other members of this family that are associated with different physical quantities and briefly consider the most recent developments related to positive geometries. Finally, we discuss the main open problems in the field. This is a Topical Review invited by Journal of Physics A: Mathematical and Theoretical.

arXiv comment47 pages

References in corpus (56)

• Direct Proof Of Tree-Level Recursion Relation In Yang-Mills Theory
• New Relations for Gauge-Theory Amplitudes
• New Recursion Relations for Tree Amplitudes of Gluons
• Gluon scattering amplitudes at strong coupling
• Scattering of Massless Particles in Arbitrary Dimension
• Dual superconformal symmetry of scattering amplitudes in N=4 super-Yang-Mills theory
• Conformal properties of four-gluon planar amplitudes and Wilson loops
• The Amplituhedron
• Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory
• A Duality For The S Matrix
• Dual Superconformal Invariance, Momentum Twistors and Grassmannians
• Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet
• Into the Amplituhedron
• Notes on the scattering amplitude / Wilson loop duality
• Positive Geometries and Canonical Forms
• Unwinding the Amplituhedron in Binary
• Exacting N=4 Superconformal Symmetry
• Positive Amplitudes In The Amplituhedron
• One-Loop Superconformal and Yangian Symmetries of Scattering Amplitudes in N=4 Super Yang-Mills
• Yangians, Grassmannians and T-duality
• The Momentum Amplituhedron
• Lectures on Yangian Symmetry
• A Proof of the Supersymmetric Correlation Function / Wilson Loop Correspondence
• A Coding of Real Null Four-Momenta into World-Sheet Co-ordinates
• Landau Singularities from the Amplituhedron
• Scattering Equations: From Projective Spaces to Tropical Grassmannians
• Towards the Amplituhedron Volume
• Cutting Deep Into The Amplituhedron
• The Correlahedron
• Stokes Polytopes : The positive geometry for ${\varphi }^4$ interactions
• 1-loop Amplitudes from the Halohedron
• The Amplituhedron and the One-loop Grassmannian Measure
• On the Positive Geometry of Conformal Field Theory
• 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 m=1 amplituhedron and cyclic hyperplane arrangements
• On Positive Geometries of Quartic Interactions II : Stokes polytopes, Lower Forms on Associahedra and Worldsheet Forms
• All-Helicity Symbol Alphabets from Unwound Amplituhedra
• Cluster Adjacency for m=2 Yangian Invariants
• The positive geometry for ${\varphi }^p$ interactions
• From Momentum Amplituhedron Boundaries to Amplitude Singularities and Back
• An Etude on Recursion Relations and Triangulations
• Amplituhedron meets Jeffrey-Kirwan Residue
• Positive Geometries and Differential Forms with Non-Logarithmic Singularities I
• Sign Flip Triangulations of the Amplituhedron
• The positive tropical Grassmannian, the hypersimplex, and the m=2 amplituhedron
• Boundaries of the Amplituhedron with amplituhedronBoundaries
• Notes on polytopes, amplitudes and boundary configurations for Grassmannian string integrals
• Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra
• Hyperbolic Geometry and Amplituhedra in 1+2 dimensions
• All-loop cuts from the Amplituhedron
• Yangian Symmetry for the Tree Amplituhedron
• On Positive Geometry and Scattering Forms for Matter Particles
• Triangulation of 2-loop MHV Amplituhedron from Sign Flips
• Emergent unitarity from the amplituhedron

Cited by in corpus (13)

• Amplitudes meet Cosmology: A (Scalar) Primer
• The SAGEX Review on Scattering Amplitudes, Chapter 7: Positive Geometry of Scattering Amplitudes
• Integrated negative geometries in ABJM
• The Loop Momentum Amplituhedron
• Momentum Amplituhedron meets Kinematic Associahedron
• Non-planar BCFW Grassmannian Geometries
• Towards Positive Geometry of Multi Scalar Field Amplitudes : Accordiohedron and Effective Field Theory
• Kleiss-Kuijf Relations from Momentum Amplituhedron Geometry
• $\mathcal{N}=7$ On-Shell Diagrams and Supergravity Amplitudes in Momentum Twistor Space
• Poles At Infinity in On-shell Diagrams
• The hypersimplex canonical forms and the momentum amplituhedron-like logarithmic forms
• Generalised wavefunction coefficients and acyclonesto-cosmohedra
• Triangular tessellations of one-loop scattering amplitudes in ${\varphi }^3$ theory