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The positive-geometry conjecture for amplituhedra

From papers

For integers satisfying ke0k e 0 and k+menk+m e n, let ZZ be a totally positive matrix and let An,k,m(Z)\mathcal{A}_{n,k,m}(Z) be the image of Gr+(k,n)\mathop{\rm Gr}_+(k,n) under the induced map. A positive geometry is a geometric space equipped with a canonical form characterized by its boundary structure. The positive-geometry conjecture. The amplituhedron An,k,m\mathcal{A}_{n,k,m} is a positive geometry. Consequently, for a positroidal triangulation C={SM}\mathcal{C}=\lbrace S_M\rbrace, its canonical form should be computed by summing the canonical forms of the images of its cells:

Ω(An,k,m)=∑SM∈CΩ(π+(SM)).\mathbf{\Omega}\left(\mathcal{A}_{n,k,m} \right)=\sum_{S_M\in\mathcal{C}} \mathbf{\Omega}\left(\pi_+(S_{M}) \right).

The positive-geometry interpretation underlies the use of canonical forms and positroidal triangulations in amplituhedron geometry. The source treats it as conjectural and gives no resolution.

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Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The positive-geometry conjecture for amplituhedra

    Let An,k,mA_{n,k,m} be the amplituhedron associated to a positive map Z:Rn→Rk+mZ:{\mathbb{R}}^n\to{\mathbb{R}}^{k+m}. A positive geometry is a pair (X,X≥0)(X,X_{\geq 0}) equipped with a canonical rational form Ω(X,X≥0)\Omega(X,X_{\geq 0}) satisfying the recursive residue axioms along its boundary components. The positive-geometry conjecture. The amplituhedron An,k,mA_{n,k,m} is a positive geometry, and the canonical form Ω(An,k,4)\Omega(A_{n,k,4}) “is” the N=4N=4 SYM amplitude. This is the central motivating conjecture of positive geometry: it seeks to explain scattering amplitudes through canonical differential forms on geometric spaces. The statement remains unresolved in the supplied context.

    source: Thomas Lam, “The combinatorial geometry of particle physics”, arXiv:2509.25372 (2025).

Sources & referencesView supporting material

Primary source

Fatemeh Mohammadi, Leonid Monin and Matteo Parisi, “Triangulations and Canonical Forms of Amplituhedra: a fiber-based approach beyond polytopes”, arXiv:2010.07254 (2021).

amplituhedracanonical-formspositive-geometryscattering-amplitudes

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