A Note on Convex Realization of Halohedron

TL;DR

This paper provides a geometric realization of the Halohedron, a polytope associated with 1-loop planar diagrams in bi-adjoint massless $\

Contribution

It constructs a polytopal realization of the Halohedron using graph cubeahedron methods, confirming its equivalence to the known realization in Big Kinematic Space.

Findings

The Halohedron can be realized as a specific graph cubeahedron.

The constructed realization matches the previously proposed one in Big Kinematic Space.

This work bridges combinatorial constructions with geometric realizations of Halohedron.

Abstract

In the recent works [arXiv:1803.05809],[arXiv: 1806.01842], Halohedron emerged as amplituhedron for 1-loop planar diagrams in bi-adjoint massless $\phi^3$ theory. Halohedron is a specific case of graph cubeahedron where the considered graph is a cycle-graph. In [arXiv: 1906.06861],[arXiv: 1501.07152], the authors provide construction of any graph cubeahedron and we use this construction to find the polytopal realization of Halohedron. We show that the Halohedron we obtain is equivalent to the proposed realization of Halohedron in `Big Kinematic Space'[arXiv: 1806.01842].