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The BCFW Tiling of the ABJM Amplituhedron

Michael Oren-Perlstein, Ran Tessler

TL;DR

The paper resolves the ABJM analog of the BCFW tiling conjecture by proving that the ABJM amplituhedron $\mathcal{O}_k(\Lambda)$ can be tiled by images of BCFW orthitroid cells, with injectivity of the mapping for positive $\Lambda$ and local separation along codimension-1 boundaries. It develops a promotion framework (Rot, Cyc, Inc, Arc moves) to inductively construct and invert the amplituhedron map on BCFW cells, using both twistors and Mandelstam variables. A key contribution is the introduction of strongly positive matrices $\mathcal{L}_k^>$, which ensure nonnegative Mandelstam variables and enable a robust tiling proof; the authors also establish local separation and a detailed boundary stratification. The results extend the geometric understanding of ABJM scattering amplitudes, generalizing ideas from the SYM amplituhedron and momentum amplituhedron while highlighting new algebraic features such as square-root structures and positivity constraints that arise in the orthogonal setting.

Abstract

The orthogonal momentum amplituhedron O_k was introduced simultaneously in 2021 by Huang, Kojima, Wen, and Zhang, and by He, Kuo, and Zhang, in the study of scattering amplitudes of ABJM theory. It was conjectured that it admits a decomposition into BCFW cells. We prove this conjecture.

The BCFW Tiling of the ABJM Amplituhedron

TL;DR

The paper resolves the ABJM analog of the BCFW tiling conjecture by proving that the ABJM amplituhedron can be tiled by images of BCFW orthitroid cells, with injectivity of the mapping for positive and local separation along codimension-1 boundaries. It develops a promotion framework (Rot, Cyc, Inc, Arc moves) to inductively construct and invert the amplituhedron map on BCFW cells, using both twistors and Mandelstam variables. A key contribution is the introduction of strongly positive matrices , which ensure nonnegative Mandelstam variables and enable a robust tiling proof; the authors also establish local separation and a detailed boundary stratification. The results extend the geometric understanding of ABJM scattering amplitudes, generalizing ideas from the SYM amplituhedron and momentum amplituhedron while highlighting new algebraic features such as square-root structures and positivity constraints that arise in the orthogonal setting.

Abstract

The orthogonal momentum amplituhedron O_k was introduced simultaneously in 2021 by Huang, Kojima, Wen, and Zhang, and by He, Kuo, and Zhang, in the study of scattering amplitudes of ABJM theory. It was conjectured that it admits a decomposition into BCFW cells. We prove this conjecture.
Paper Structure (62 sections, 137 theorems, 135 equations, 23 figures)

This paper contains 62 sections, 137 theorems, 135 equations, 23 figures.

Table of Contents

  1. Introduction
  2. A Speed of Light Review of Positive Grassmannians, the Positive Orthogonal Grassmannian and the ABJM Amplituhedron
  3. Tilings and BCFW Tilings
  4. Main Results
  5. Relation with Existing Literature
  6. Plan of the Paper
  7. BCFW Graphs and the ABJM Amplituhedron
  8. BCFW Graphs
  9. The ABJM Amplituhedron and Its Natural Coordinates
  10. Twistors Variables
  11. Promotion
  12. The Basic Moves
  13. The Rot Move
  14. The Cyc Move
  15. The Inc Move
  16. ...and 47 more sections

Key Result

Theorem 1.3.1

For every positive $\Lambda\in \mathrm{Mat}^>_{2k \times (k+2)}$ the BCFW cells $\Omega\in\mathrm{BCFW}_k$ map injectively to the amplituhedron. Moreover, two BCFW cells whose closures share a common codimension $1$ boundary are locally separated near that boundary.

Figures (23)

  • Figure 1: a BCFW graph with its triangle tree superimposed in blue
  • Figure 2: building OG graphs using the $\mathrm{Arc}$ move (the added arcs for each step are in blue)
  • Figure 3: The effect of $\mathrm{Arc}_{4,2i}$ on the triangle graph
  • Figure 4: opening a trigonometric or hyperbolic oriented vertex
  • Figure 5: the BCFW graph $\Gamma$ with internal vertex $v$ that is adjacent to an external vertex
  • ...and 18 more figures

Theorems & Definitions (327)

  • Definition 1.2.1
  • Theorem 1.3.1
  • Theorem 1.3.2
  • Definition 2.1.1: BCFWrec, Section 4.4
  • Definition 2.1.2
  • Proposition 2.1.3: companion
  • Definition 2.1.4
  • Theorem 2.1.5: PosOG, Section 4.4, companion
  • Proposition 2.1.6: companion
  • Proposition 2.1.7: companion
  • ...and 317 more