Positive Geometries: Structure & Scattering

• Positive geometries are mathematical structures defined by a complex variety, a closed positive region, and a canonical form that satisfies a recursive residue property.
• They can be constructed through methods like triangulation of convex polytopes and the positive Grassmannian, linking discrete combinatorics with continuous geometry.
• Their unique canonical forms, with simple poles and residue recursion, play a central role in encoding scattering amplitudes in high-energy theoretical physics.

A positive geometry is a rich mathematical structure connecting algebraic geometry, combinatorics, and high-energy theoretical physics, particularly the theory of scattering amplitudes. It is defined as a triple (X, X_{≥0}, Ω), where X is a complex (quasi-)projective variety of dimension n, X_{≥0} ⊂ X(ℝ) is a closed semi-algebraic “positive region” of real points (with boundary ∂X_{≥0}), and Ω(X_{≥0}) is a meromorphic top form on X, referred to as the canonical form. This structure is characterized by a precise recursion: Ω has only simple poles along the irreducible components of ∂X_{≥0}, and for each such boundary component D, the residue Res_{D}(Ω) is, up to sign, the canonical form of the corresponding lower-dimensional positive geometry (D, D_{≥0}). On zero-dimensional geometries, Ω is ±1. The uniqueness and explicit recursive construction of Ω, its deep connection to combinatorics, and its role in encoding the analytic and topological features of physical amplitudes make positive geometry a central object in contemporary mathematical physics (Ranestad et al., 18 Feb 2025, Eur et al., 28 Feb 2025, Brown et al., 6 Jan 2025, Arkani-Hamed et al., 2017, Moerman, 2023, Shen et al., 10 Mar 2026).

1. Formal Definition and Recursive Structure

A positive geometry is defined by three data:

1. X: a complex projective (or quasi-projective) variety of dimension n, typically defined over ℝ.
2. X_{≥0}: a closed semi-algebraic subset X_{≥0} ⊂ X(ℝ) (“positive region”) whose boundary ∂X_{≥0} is a union of hypersurfaces in X. The interior X_{>0} is an oriented real n-manifold.
3. Ω(X_{≥0}): a unique (up to scale) rational n-form on X, with the following properties:
   • Ω has only simple poles on the irreducible components of ∂X_{≥0}, no other poles.
   • For any boundary divisor D ⊂ ∂X_{≥0}, the residue

   $\operatorname{Res}_{D}(\Omega) = \Omega(D_{≥0}),$

   where (D, D_{≥0}) is itself a positive geometry of dimension n−1. - For zero-dimensional geometries (points), Ω=±1.

This recursive structure ensures that each face of the positive region inherits, together with its canonical residue, the structure of a positive geometry of one lower dimension. The canonical form Ω(X_{≥0}) is uniquely determined by these data and conditions; it is rational, with logarithmic singularities on the boundary and fixed leading residues (Ranestad et al., 18 Feb 2025, Arkani-Hamed et al., 2017).

2. Canonical Forms, Uniqueness, and Computation

The canonical form Ω(X_{≥0}) is the central invariant of a positive geometry. Its defining features:

• Rational Top-Form: Ω is a meromorphic n-form on X, characterized by its simple poles and residue recursion.
• Uniqueness: If two rational forms satisfy the pole and residue property, their difference is a holomorphic top-form on a projective variety and must vanish—therefore, the canonical form is unique up to sign (Eur et al., 28 Feb 2025, Arkani-Hamed et al., 2017).
• Residue Recursion: The entire structure of Ω is determined by the unit leading residues on all codimension-n faces (points).
• Explicit Construction:
  • For polytopes cut out by linear forms {ℓ_j}, Ω can often be written as a sum over permutations or bases, with denominators encoding the facets,

  $$\Omega_P = \sum_{I\subset\{1,\dots,k\}, |I|=n} \det(\ell_i(e_j))_{i\in I} \cdot \frac{dx_1 \wedge \cdots \wedge dx_n}{\prod_{i\in I} \ell_i(x)}$$

  (Eur et al., 28 Feb 2025). - For convex polytopes or more general settings, the canonical form can be constructed using triangulations, push-forward under morphisms, or volume duality methods such as Brion or Duistermaat–Heckman formulas (Arkani-Hamed et al., 2017).

The explicit recursive behavior of residues codifies a rigorous connection between the combinatorial structure of the boundary and the analytic structure of Ω.

3. Prototypical Examples and Key Constructions

Positive geometries appear in multiple settings unified by this recursive structure.

A. Convex Polytopes:

For P ⊂ ℝ^n, a full-dimensional polytope, with facets given by supporting hyperplanes, the canonical form has simple poles on all facets, and residues along facets reproduce the canonical form of the lower-dimensional facet (Ranestad et al., 18 Feb 2025). The explicit realization includes classical polytopes, cyclic polytopes, and amplituhedra in particular limits.

B. Positive Grassmannian and Positroids:

The totally nonnegative Grassmannian Gr(k, n)_{≥0} is the locus where all Plücker coordinates are nonnegative. Its positroid stratification yields a cell decomposition into semi-algebraic regions characterized by specific vanishing/nonvanishing patterns of minors. Each positroid cell is a positive geometry, with a canonical differential form with simple poles along the vanishing Plückers and residue recursion matching the lower strata (Ranestad et al., 18 Feb 2025, Arkani-Hamed et al., 2017).

C. Amplituhedron:

For a fixed positive matrix Z, the image of the positive Grassmannian Gr(k, n){≥0} under the map [X] ↦ [X·Z] into Gr(k, k+m) yields the amplituhedron 𝒜{n,k,m}(Z). Conjecturally, at each tree level and for many loop levels (with possible modifications), 𝒜_{n,k,m}(Z) is a positive geometry with a unique canonical form, whose boundaries are unions of Schubert-type divisors corresponding to cyclically adjacent vanishing minors (Ranestad et al., 18 Feb 2025, Stalknecht, 2024).

D. Moduli Spaces and Binary Geometries:

The moduli space M_{0,n} of n marked points on ℙ^1 carries a positive locus with canonical form the Parke–Taylor form. Such “binary” geometries can be explicitly described via cross-ratio coordinates and binomial constraints, providing explicit connections to the CHY formalism for scattering amplitudes (Ranestad et al., 18 Feb 2025).

E. Oriented Matroids:

Topes of an oriented matroid, generalizing chambers of real hyperplane arrangements, admit canonical forms as elements of the reduced Orlik–Solomon algebra. These forms satisfy the same polar and residue recursion mirroring the boundary complex of the matroid cell (Eur et al., 28 Feb 2025).

F. Cluster, Toric, and Tropical Varieties:

Cluster varieties, toric varieties, and their positive loci with log-form volume elements are extended examples, showing the breadth of the framework (Arkani-Hamed et al., 2017).

4. Applications to Scattering Amplitudes and Mathematical Physics

Positive geometries arise in the modern theory of scattering amplitudes as geometric domains whose canonical forms directly encode physical amplitudes:

• Bi-Adjoint Scalar Theory: The kinematic associahedron (ABHY construction) serves as a positive geometry, with canonical form reproducing tree-level S-matrix elements. Its boundaries correspond to factorization channels of amplitudes (Moerman, 2023, Ranestad et al., 18 Feb 2025).
• N=4 SYM and Amplituhedron: The amplituhedron encodes planar N=4 SYM amplitudes, with the canonical form corresponding to the integrand. The stratification by positroid cells mirrors the pole/zero structure of the amplitude (Ranestad et al., 18 Feb 2025, Stalknecht, 2024, Arkani-Hamed et al., 2017, Shen et al., 10 Mar 2026).
• Push-Forward and CHY Formalism: The connection to the CHY framework is realized via push-forward of canonical forms along the scattering equations from the moduli space of marked points to kinematic space, relating the mathematical definition to explicit analytic formulas for amplitudes (Lukowski et al., 2022, Stalknecht, 2024).
• Generalized Theories: Deformations (e.g., via scaling and shifting in kinematic variables) realize positive geometries for a wide spectrum of physical models, including scalar field theories with massive components or polynomial interactions (Jagadale et al., 2022, Aneesh et al., 2019, Jagadale et al., 2023).
• Stringy Generalizations: Infinite unions of polytopal regions (associahedral grids) produce positive geometries whose canonical forms have infinitely many poles, modeling the tower of resonances in string amplitudes and yielding a geometric interpretation of KLT kernels and string-theory phenomena (Bartsch et al., 27 Aug 2025, Kim et al., 30 Mar 2026).

5. Open Problems and Research Directions

Several central directions define the future landscape of positive geometry:

• Universal Cell Decomposition: The search for full (positroid-like) cell decompositions for all positive geometries, including amplituhedra and positive loci in orthogonal Grassmannians, remains open (Ranestad et al., 18 Feb 2025).
• Weighted and Higher-Genus Positive Geometries: The generalization to loop-level geometries, where the canonical form may require non-trivial weights on facets, and to higher-genus boundaries (e.g., amplituhedra with elliptic or non-logarithmic structures), is ongoing (Ranestad et al., 18 Feb 2025, Brown et al., 6 Jan 2025).
• Scheme-Theoretic Boundary and Tropicalization: Determining the scheme structure of positive regions and their boundaries, particularly for moduli spaces under degenerations, reciprocal varieties, and tropical compactifications, is an active area (Ranestad et al., 18 Feb 2025, Early et al., 2023).
• Algebraic and Combinatorial Classification: There is an ongoing effort to classify all positive geometries in connection with oriented matroids, hyperplane arrangements, Dressians, and tropical Grassmannians, including determining the precise relationship between combinatorial invariants and the canonical forms (Eur et al., 28 Feb 2025, Ranestad et al., 18 Feb 2025).
• Extension to Singular and Weighted Cases: The framework is expanding to include generalized positive geometries where log singularities appear in boundary interiors (e.g., Vandermonde cells with cuspidal cubics) and where canonical forms may incorporate non-logarithmic singularities relevant to cosmology and gravity (Mohammadi et al., 13 Oct 2025, Benincasa et al., 2020).
• Interplay with Analysis and D-Module Theory: Canonical forms satisfy holonomic D-module structures (e.g., GKZ systems), and analysis of their asymptotic behavior (e.g., via saddle-point methods) provides additional structure (Ranestad et al., 18 Feb 2025).

6. Role in Broader Mathematics and Concluding Remarks

Positive geometry synthesizes real, complex, and tropical aspects of algebraic geometry with analysis and combinatorics, providing canonical forms as the bridge between geometry and analytic invariants such as periods, integrands, and amplitudes. Its recursion by boundary residues reflects and mathematically underpins principles of factorization and locality in quantum field theory. The robust functoriality of canonical forms—under pushforwards, blow-ups, and products—connects it deeply with mixed Hodge theory and the topology of moduli spaces (Brown et al., 6 Jan 2025, Sturmfels et al., 12 May 2026).

Key open challenges include a systematic classification of all positive geometries (especially non-polytopal ones such as amplituhedra and cluster varieties of high genus), a combinatorial understanding of their boundary posets, and explicit computational frameworks for canonical forms in general algebraic settings (Ranestad et al., 18 Feb 2025, Brown et al., 6 Jan 2025, Early et al., 2023).

Positive geometry has thus become a nexus for cross-disciplinary research, unifying structures in pure mathematics and advancing the formalism of quantum field theory scattering (Ranestad et al., 18 Feb 2025, Arkani-Hamed et al., 2017, Bartsch et al., 27 Aug 2025).