The Amplituhedron

TL;DR

This work proposes the amplituhedron as a positive-geometry framework that computes planar ${\cal N}=4$ SYM scattering amplitudes, replacing traditional locality and unitarity with positivity-derived constraints. It builds the tree amplituhedron ${\cal A}_{n,k}(Z)$ via $Y=CZ$ with $C\in G_+(k,n)$ and positive external data, establishing a cell decomposition and a canonical form whose integral yields the superamplitude, with Yangian symmetry emerging naturally. It extends to loops through the loop amplituhedron ${\cal A}_{n,k,L}(Z)$ and a loop form $\Omega_{n,k,L}$, showing locality and unitarity arise from positivity, and demonstrates simple all-loop structures in the four-particle case. The paper also outlines a master amplituhedron and broader generalizations, pointing toward connections with integrability and potential extensions beyond planar ${\cal N}=4$ SYM to other theories and quantum-gravity contexts.

Abstract

Perturbative scattering amplitudes in gauge theories have remarkable simplicity and hidden infinite dimensional symmetries that are completely obscured in the conventional formulation of field theory using Feynman diagrams. This suggests the existence of a new understanding for scattering amplitudes where locality and unitarity do not play a central role but are derived consequences from a different starting point. In this note we provide such an understanding for N=4 SYM scattering amplitudes in the planar limit, which we identify as ``the volume" of a new mathematical object--the Amplituhedron--generalizing the positive Grassmannian. Locality and unitarity emerge hand-in-hand from positive geometry.