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The Amplituhedron BCFW Triangulation

Chaim Even-Zohar, Tsviqa Lakrec, Ran J. Tessler

TL;DR

This work proves that the Amplituhedron $\\\mathcal{A}_{n,k,4}$ admits a BCFW-type triangulation by images of positroid cells $\\mathcal{BCFW}_{n,k}$, providing a full combinatorial-accounting framework based on chord diagrams, domino matrices, and decorated permutations. It develops a robust set of tools—matrix-embeddings, a constructive domino-matrix algorithm, and twistors/functionaries with promotion—to establish injectivity, separation, and a precise boundary description for the BCFW cells, culminating in a triangulation where the interior is homeomorphic to an open ball. The work also shows the equivalence of multiple representations (chord diagrams, lattice walks, $$-diagrams, pipe dreams) and proves that domino-matrix Representatives characterize the corresponding BCFW positroid cells with a unique preimage under the amplituhedron map. These results provide a scalable, structure-rich blueprint for extending BCFW-type triangulations to other $m$ and triangulation schemes, with potential impact on understanding scattering amplitudes via positive geometries.

Abstract

The amplituhedron Ank4 is a geometric object, introduced by Arkani-Hamed and Trnka (2013) in the study of scattering amplitudes in quantum field theories. They conjecture that Ank4 admits a decomposition into images of BCFW positroid cells, arising from the Britto--Cachazo--Feng--Witten recurrence (2005). We prove that this conjecture is true.

The Amplituhedron BCFW Triangulation

TL;DR

This work proves that the Amplituhedron admits a BCFW-type triangulation by images of positroid cells , providing a full combinatorial-accounting framework based on chord diagrams, domino matrices, and decorated permutations. It develops a robust set of tools—matrix-embeddings, a constructive domino-matrix algorithm, and twistors/functionaries with promotion—to establish injectivity, separation, and a precise boundary description for the BCFW cells, culminating in a triangulation where the interior is homeomorphic to an open ball. The work also shows the equivalence of multiple representations (chord diagrams, lattice walks, -diagrams, pipe dreams) and proves that domino-matrix Representatives characterize the corresponding BCFW positroid cells with a unique preimage under the amplituhedron map. These results provide a scalable, structure-rich blueprint for extending BCFW-type triangulations to other and triangulation schemes, with potential impact on understanding scattering amplitudes via positive geometries.

Abstract

The amplituhedron Ank4 is a geometric object, introduced by Arkani-Hamed and Trnka (2013) in the study of scattering amplitudes in quantum field theories. They conjecture that Ank4 admits a decomposition into images of BCFW positroid cells, arising from the Britto--Cachazo--Feng--Witten recurrence (2005). We prove that this conjecture is true.
Paper Structure (27 sections, 75 theorems, 316 equations, 3 algorithms)

This paper contains 27 sections, 75 theorems, 316 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Definitions for BCFW Cells
  3. Chord Diagrams
  4. Domino Matrices
  5. BCFW Cells in the Combinatorial Literature
  6. BCFW Cells from Chord Diagrams
  7. Example
  8. Structure of BCFW Cells
  9. Matrix Operations
  10. Generating Domino Matrices
  11. The Generated Cell
  12. Agreement of Permutations
  13. Another Algorithm
  14. The Domino Theorem
  15. Twistors and Functionaries
  16. ...and 12 more sections

Key Result

Theorem 1.3

The map $S \to \widetilde{Z}(S)$ is injective for every cell $S \in \mathcal{BCFW}_{n,k}$.

Theorems & Definitions (235)

  • Definition 1.1: Bao and He bao2019m
  • Conjecture 1.2: Arkani-Hamed and Trnka arkani2014amplituhedron
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Definition 2.4
  • ...and 225 more