The two-loop Amplituhedron

TL;DR

This paper explores the geometric structure of the two-loop four-point Amplituhedron, extending previous work on the one-loop case to understand its algebraic and stratification properties.

Contribution

It provides the first detailed analysis of the two-loop four-point Amplituhedron's geometry, building on prior one-loop results.

Findings

Characterization of the two-loop four-point Amplituhedron's stratification.

Analysis of its algebraic structure and face arrangements.

Extension of known one-loop properties to the two-loop case.

Abstract

The loop-Amplituhedron $\mathcal{A}^{(L)}_{n}$ is a semialgebraic set in the product of Grassmannians $\mathrm{Gr}_{\mathbb{R}}(2,4)^L$. Recently, many aspects of this geometry for the case of $L=1$ have been elucidated, such as its algebraic and face stratification, its residual arrangement and the existence and uniqueness of the adjoint. This paper extends this analysis to the simplest higher loop case given by the two-loop four-point Amplituhedron $\mathcal{A}^{(2)}_4$.