Kinematic Stratifications
Veronica Calvo Cortes, Hadleigh Frost, Bernd Sturmfels
TL;DR
This paper studies the stratification of Mandelstam regions, the Gram-geometry of momentum vectors in Minkowski space, by signs of matrix entries and by rank-two matroids, with emphasis on massless scattering and momentum conservation. It develops a semialgebraic description of Mandelstam regions via principal-minor sign conditions, shows the Lorentzian region corresponds to nonnegative Gram matrices, and introduces a signed rank-two matroid framework to classify massless strata, including explicit dimension counts and combinatorial formulas. Topologically, the authors show that stratum closures are governed by a matroid poset and that each stratum is homotopy equivalent to an orbit configuration space on spheres (reaching moduli spaces $M_{0,m}(\mathbb{C})$ in 4D), revealing deep links between kinematics, algebraic geometry, and topology. In the momentum-conserving massless case (MMC), they characterize which signed matroids occur and provide dimension and counting formulas, with illuminating low-dimensional examples that tie to classical objects like the Igusa quartic; the results lay groundwork for extensions to massive cases and non-Mandelstam Gram matrices. Overall, the work bridges scattering amplitudes with geometric combinatorics and configuration-space topology, offering a structured, lattice-based view of kinematic regions and their singularities.
Abstract
We study stratifications of regions in the space of symmetric matrices. Their points are Mandelstam matrices for momentum vectors in particle physics. Kinematic strata in these regions are indexed by signs and rank two matroids. Matroid strata of Lorentzian quadratic forms arise when all signs are non-negative. We characterize the posets of strata, for massless and massive particles, with and without momentum conservation.