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Positive Geometry Program

Updated 24 September 2025
  • Positive Geometry Program is an interdisciplinary framework that combines algebraic, combinatorial, tropical geometry, and theoretical physics to encode physical observables through unique canonical forms.
  • It leverages geometric structures such as associahedra, accordiohedra, and moduli spaces, enabling the computation of scattering amplitudes through recursive residue and triangulation techniques.
  • The program provides practical methodologies for amplitude calculations by mapping combinatorial mutations and dual constructions to well-defined canonical forms that satisfy locality and unitarity.

The Positive Geometry Program is an interdisciplinary research enterprise combining elements from algebraic geometry, combinatorics, tropical geometry, and theoretical physics. It aims to encode physical observables—most notably, scattering amplitudes and correlators—through a geometric framework, specifically via classes of spaces called positive geometries equipped with canonical differential forms. This viewpoint has transformed the computation and conceptual understanding of amplitudes in various quantum field theories, as well as inspired new methods in geometric modeling and related disciplines.

1. Foundations and Definition of Positive Geometry

A positive geometry is defined as a pair (X,X0)(X, X_{\geq 0}), where XX is a complex algebraic variety and X0X_{\geq 0} is a closed semialgebraic subset of its real part with oriented interior. Each positive geometry possesses a unique canonical meromorphic top-form Ω(X,X0)\Omega(X, X_{\geq 0}) that has simple poles only along the boundary of X0X_{\geq 0} and satisfies a recursive residue property: the residue of the canonical form on any boundary component CC is the canonical form of CC itself (Lam, 2022, Vaitkus, 2021, Ranestad et al., 18 Feb 2025). Fundamental examples include intervals, polytopes (e.g., the associahedron), moduli spaces (e.g., M0,nM_{0,n}), Grassmannians, flag varieties, and their positive parts. The canonical form has key properties:

  • Recursive boundary structure: The residue on any facet yields the canonical form for that facet.
  • Additivity: Subdivisions (“triangulations”) of X0X_{\geq 0} correspond to summing canonical forms.
  • Duality: Canonical forms are often represented as volumes or Laplace transforms over dual objects (Lam, 2022, Telen, 5 Jun 2025).
  • Invariance and topology: Positive regions are often homeomorphic to closed balls; top cohomology is generally one-dimensional.

In projective coordinates, canonical forms take the shape Ω(X,X0)=C(x)ω(x)\Omega(X, X_{\geq 0}) = C(x)\omega(x), with C(x)C(x) rational and ω(x)\omega(x) the top wedge product (Vaitkus, 2021).

2. Applications to Scattering Amplitudes

The Positive Geometry Program originated in particle physics, where Arkani-Hamed, Bai, and Lam, among others, discovered that various amplitudes could be encoded as residues of canonical forms on positive geometries (Arkani-Hamed et al., 2017, Banerjee et al., 2018, Raman, 2019, Ranestad et al., 18 Feb 2025). Key examples include:

  • ϕ3\phi^3 theory: Planar tree-level amplitudes are captured by the associahedron sitting in kinematic space, with the amplitude given by the residue of its canonical form. Planar Feynman diagrams correspond to triangulations of an nn-gon.
  • ϕ4\phi^4 theory: Amplitudes are encoded using Stokes polytopes, associated to quadrangulations, but a single polytope does not suffice; the amplitude is a weighted sum over all polytopes with weights determined by combinatorial equivalence classes (“primitive quadrangulations”) (Banerjee et al., 2018).
  • ϕp\phi^p interactions (for p>4p>4): The generalization uses “accordiohedra,” where p-angulations and their Q-compatibility structure yields a family of positive geometries. The amplitude is a weighted sum over all primitive accordiohedra (Raman, 2019).
  • String and mixed amplitudes: The “associahedral grid” provides a geometric realization of the inverse KLT kernel, reproducing stringy features like infinite resonance structure and the α\alpha'-dependence, unifying bi-adjoint ϕ3\phi^3 and NLSM pion amplitudes (Bartsch et al., 27 Aug 2025).

All such amplitudes are computed by pulling back universal “scattering forms” onto the positive geometry and extracting residues.

3. Combinatorics, Canonical Forms, and Factorization

Combinatorics play a central role in determining both the geometry and the associated amplitudes:

  • Triangulations/Angulations: The structure of Feynman diagrams is encoded as (tri-, quad-, p-)angulations of an nn-gon.
  • Mutations/flips: Moves between diagrams are characterized by specific combinatorial mutations (e.g., flips for triangulations/quadrangulations).
  • Residue structure and locality/unitarity: Boundary factorization properties guarantee that the residues correspond to the expected physical consistency conditions, such as unitarity (factorization into lower-point amplitudes) and locality (Banerjee et al., 2018, Raman, 2019).
  • Weight prescription: The correct amplitude requires assigning combinatorial weights so that each term appears with unit residue, as dictated by the Fuss–Catalan enumeration and cyclic symmetry classes.

Canonical forms (differential forms with logarithmic singularities) naturally encode the physical singularities of amplitudes, and their additivity allows for elegant triangulation and decomposition formulas (Lam, 2022, Vaitkus, 2021).

4. Extensions: Moduli Spaces, Stringy and Tropical Geometry

Beyond polygons and standard polytopes, positive geometries appear in a wide range of algebraic and tropical contexts:

  • Moduli spaces M0,nM_{0,n}: Treated as positive and binary geometries; canonical Parke–Taylor forms describe both the combinatorics and boundary structure, relating to the associahedron and dihedral/binary coordinates (Lam, 27 May 2024).
  • Del Pezzo moduli and Weyl group symmetry: Cell decompositions of these moduli spaces (very affine varieties) lead to “pezzotopes” whose combinatorics are controlled by root systems and Weyl group actions, and whose canonical forms are constructed via solutions to “u-equations” and likelihood (scattering) equations (Early et al., 2023).
  • Tropicalization: String amplitudes and integral representations are often analyzed by tropicalizing potentials, giving rise to polyhedral fans whose domains of linearity correspond to quantum field theory limits (Feynman diagram combinatorics) (Lam, 27 May 2024).
  • Generalization to barycentric interpolation and computer graphics: Canonical forms provide a conceptual framework for generalized barycentric coordinates, Wachspress coordinates, mean-value coordinates, and their projective/tropical deformations (Vaitkus, 2021).

5. Loop-Level and Dual Positive Geometries

Recent research has begun exploring positive geometry at loop level and dual constructions:

  • Local positive spaces and “Amplituhedron-Prime”: For one-loop MHV amplitudes in planar SYM, the geometry is defined by sign-flip conditions, yielding “local positive spaces” and “Amplituhedron-Prime,” which differ in their bulk structure from the standard Amplituhedron but share the same boundary and canonical form (Herrmann et al., 2020).
  • Triangulation by chiral pentagons/octagons: Specific geometric structures (chiral pentagons, octagons) serve as natural local building blocks for loop amplitudes, and dualization procedures show that these blocks internally triangulate dual Amplituhedron spaces.
  • Pushforward construction: The canonical forms of complicated positive geometries can often be realized as pushforwards of standard forms (e.g., from a simplex or toric variety), connecting to toric and moment map geometry (Lam, 2022).

6. Mathematical Challenges and Open Questions

The Positive Geometry Program, while conceptually robust, faces several open challenges (Ranestad et al., 18 Feb 2025):

  • Existence and uniqueness: Rigorous construction of the canonical form on a given positive geometry, especially for general amplituhedra or at higher loop levels.
  • Algorithmic computation: Efficient solution of scattering or likelihood equations for increasingly complex geometries and for moduli spaces with intricate combinatorics.
  • Interplay between real, complex, and tropical geometry: Merging insights from these domains to reconcile compactification, boundary arrangements, and algebraic/tropical fans.
  • Differential equations/D-modules: Systematic application of D-module theory and GKZ systems to analyze amplitude integrals, canonical forms, and their analytic continuation.
  • Generalization: Application to broader classes of amplitudes (e.g., gravity, mixed correlators, loop integrals) and investigation into binary and Pellspace geometries.

A plausible implication is that further development of triangulation and residue techniques will clarify the uniqueness and analytic properties of canonical forms for more intricate positive geometries.

Progress in the Positive Geometry Program is rapid, with substantial interdisciplinary impact:

  • Unification of amplitude computation: Positive geometry offers a rigorous and geometric language underlying combinatorics and residue calculus for amplitudes in quantum field theory and string theory (Bartsch et al., 27 Aug 2025).
  • Connections to algebraic statistics and optimization: Convex positive regions also model feasible sets and probability simplices.
  • Applications to geometric modeling: Barycentric interpolation, projective coordinates, and splines may be generalized using canonical forms (Vaitkus, 2021).
  • Mixed Hodge/D-module structures: Recent work incorporating Hodge-theoretic aspects indicates a broader mathematical framework (Telen, 5 Jun 2025).
  • Educational outreach and open problems: Lecture notes and comprehensive expositions are fostering wider adoption and posing new challenges for mathematicians and physicists (Telen, 5 Jun 2025, Ranestad et al., 18 Feb 2025).

The field is poised to provide further foundational tools for both physics and mathematics by explicating the interplay between geometric, combinatorial, and analytic structures implicit in physical theories.

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