The combinatorial geometry of particle physics

TL;DR

The paper surveys how scattering amplitudes in perturbative quantum field theory are naturally realized as canonical forms on positive geometries, unifying diverse mathematical structures from polytopes to Grassmannians, moduli spaces, hyperplane arrangements, matroids, and tropical varieties. It develops and motivates the amplituhedron and positroid geometry as geometric embodiments of planar, supersymmetric theories, and shows how positivity and locality constrain amplitude structure. Key contributions include formalizing positive geometries, linking on-shell diagrams to positroid varieties, and extending amplitude concepts to matroids via twisted cohomology and Orlik–Solomon algebras, with tropical and moduli-space connections that recast string-theoretic limits in geometric terms. The significance lies in providing a unifying, geometry-centric language that may simplify calculations and reveal deep symmetries in scattering processes, with promising directions for matroid amplitudes and beyond.

Abstract

Recent breakthroughs in the study of scattering amplitudes have uncovered profound and unexpected connections with combinatorial geometry. These connections range from classical structures -- such as polytopes, matroids, and Grassmannians -- to more modern developments including positroid varieties and the amplituhedron. Together they point toward the unifying framework of positive geometry, in which geometric domains canonically determine analytic functions governing scattering processes. This survey traces the emergence of positive geometry from the physics of amplitudes, building towards recent progress on amplitudes for matroids.

Figures (6)

• Figure 2.1: A cubic planar tree with six leaves. Each internal edge is labeled with the momentum of the particle traveling along it, which can be calculated by momentum conservation.
• Figure 4.1: Left: the Feynman box diagram has four incoming particles with momenta ${\mathbf{p}}_1,{\mathbf{p}}_2,{\mathbf{p}}_3,{\mathbf{p}}_4$. Center: drawing the dual planar graph, we assign dual momenta to the vertices so that the differences along edges are momenta in the left diagram. Right: we put all the original momenta on-shell, and replace dual momenta with lines $L = \varphi({\mathbf{y}})$ labeling the vertices. Edges in this dual graph represent intersecting lines and are labeled by the point of intersection.
• Figure 5.1: Left: an on-shell diagram, or plabic graph, where boundary vertices are assumed to be white. Center: a vector-relation configuration consists of vectors $W_w$ assigned to each white vertex $w$, including the boundary vertices. Right: the trip permutation sends $1$ to $4$ by turning left (right) at white (black) vertices.
• Figure 8.1: Left: The $2$-dimensional associahedron $\mathop{\mathrm{Ass}}\nolimits_2$ is a pentagon. The vertices have been labeled by the five cubic planar trees with five leaves. Right: The associahedron in \ref{['thm:GLX']} is a Minkowski sum (we do not depict the $0$-simplices which give a translation of the polytope). We have projected the picture to the plane by omitting the second coordinate (corresponding to the basis vector $\epsilon_2$).
• Figure 9.1: Left: the scattering of closed strings ($S^1$) is represented by a point on ${\mathcal{M}}_{2,4}({\mathbb{C}})$, where marked points, or punctures, are drawn as cusps at infinity. Center: the scattering of open strings ($[0,1]$) is represented by a point on ${\mathcal{M}}_{0,5}({\mathbb{R}})$. Right: as the string length goes to 0, we obtain a Feynman diagram.

Theorems & Definitions (28)

• Example 2.1
• Proposition 4.1
• Proof 1
• Definition 4.2
• Definition 5.1
• Definition 5.2
• Theorem 5.3: AGPRLamCDM
• Example 5.4
• Example 5.5
• Definition 6.1