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.

47 pages

References in corpus (11)

• New Relations for Gauge-Theory Amplitudes
• Gluon scattering amplitudes at strong coupling
• Conformal properties of four-gluon planar amplitudes and Wilson loops
• Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory
• Notes on the scattering amplitude / Wilson loop duality
• Exacting N=4 Superconformal Symmetry
• One-Loop Superconformal and Yangian Symmetries of Scattering Amplitudes in N=4 Super Yang-Mills
• Lectures on Yangian Symmetry
• Scattering Equations: From Projective Spaces to Tropical Grassmannians
• Positive Geometries and Differential Forms with Non-Logarithmic Singularities I
• All-loop cuts 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