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Splitting CEGM Amplitudes

Bruno Giménez Umbert, Bernd Sturmfels

TL;DR

This work advances the understanding of CEGM amplitudes by introducing split kinematics ${\cal K}^{\textrm{split}}$, on which $m_n^{(k)}$ factorizes into simplex amplitudes and reveals zeros, linking stringy $\alpha'\to0$ limits to restricted biadjoint scalars. It develops a positive parametrization of the moduli spaces $X(k,n)$ via a network representation, and demonstrates that split amplitudes can be obtained through a global Schwinger formula as a Laplace transform over tropical determinantal varieties, connecting to tropical geometry and matroid amplitudes. The paper then extends these ideas to determinantal varieties of rank-two matrices, showing that bicolored trees and Bergman fans provide a new class of amplitudes consistent with Lam's theory and offering a path to generalizations involving higher-rank tropical geometries. Overall, the framework unifies stringy, tropical, and combinatorial perspectives to derive and analyze CEGM amplitudes on split loci, with potential applications to broader determinantal and hyperplane tropical geometries.

Abstract

The CEGM formalism offers a general framework for scattering amplitudes, which rests on Grassmannians, moduli spaces and tropical geometry. The physical implications of this generalization are still to be understood. Conventional wisdom says that key features of scattering amplitudes, like factorization at their poles into lower-point amplitudes, are associated to their singularities. The factorization behavior of CEGM amplitudes at their poles is interesting but complicated. Recent developments have revealed important properties of standard particle and string scattering amplitudes from factorizations, known as splits, that happen away from poles. In this paper we introduce a kinematic subspace on which the CEGM amplitude splits into very simple rational functions. These functions, called simplex amplitudes, arise from stringy integrals for the multivariate beta function, and also from restricting the biadjoint scalar amplitude in quantum field theory to certain kinematic loci. Using split kinematics we also discover a specific class of zeros of the CEGM amplitude. Our construction rests on viewing positive moduli space as a product of simplices, and it suggests a novel approach for deriving scattering amplitudes from tropical determinantal varieties.

Splitting CEGM Amplitudes

TL;DR

This work advances the understanding of CEGM amplitudes by introducing split kinematics , on which factorizes into simplex amplitudes and reveals zeros, linking stringy limits to restricted biadjoint scalars. It develops a positive parametrization of the moduli spaces via a network representation, and demonstrates that split amplitudes can be obtained through a global Schwinger formula as a Laplace transform over tropical determinantal varieties, connecting to tropical geometry and matroid amplitudes. The paper then extends these ideas to determinantal varieties of rank-two matrices, showing that bicolored trees and Bergman fans provide a new class of amplitudes consistent with Lam's theory and offering a path to generalizations involving higher-rank tropical geometries. Overall, the framework unifies stringy, tropical, and combinatorial perspectives to derive and analyze CEGM amplitudes on split loci, with potential applications to broader determinantal and hyperplane tropical geometries.

Abstract

The CEGM formalism offers a general framework for scattering amplitudes, which rests on Grassmannians, moduli spaces and tropical geometry. The physical implications of this generalization are still to be understood. Conventional wisdom says that key features of scattering amplitudes, like factorization at their poles into lower-point amplitudes, are associated to their singularities. The factorization behavior of CEGM amplitudes at their poles is interesting but complicated. Recent developments have revealed important properties of standard particle and string scattering amplitudes from factorizations, known as splits, that happen away from poles. In this paper we introduce a kinematic subspace on which the CEGM amplitude splits into very simple rational functions. These functions, called simplex amplitudes, arise from stringy integrals for the multivariate beta function, and also from restricting the biadjoint scalar amplitude in quantum field theory to certain kinematic loci. Using split kinematics we also discover a specific class of zeros of the CEGM amplitude. Our construction rests on viewing positive moduli space as a product of simplices, and it suggests a novel approach for deriving scattering amplitudes from tropical determinantal varieties.
Paper Structure (7 sections, 1 theorem, 62 equations, 3 figures)

This paper contains 7 sections, 1 theorem, 62 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Simplex Amplitudes
  3. Positive Parametrization
  4. CEGM Revisited
  5. Split Kinematics
  6. Global Schwinger Formula
  7. Amplitudes from Determinantal Varieties

Key Result

Proposition 7.1

The map (eq:GrMap) is surjective. Over a rank $\,2$ matrix $X \in \mathcal{D}^{p \times q}_2$, the fiber of (eq:GrMap) is the curve consisting of all matrices where $t \in \mathbb{P}^1$. Here, $P \in G(2,p)$ is recovered as the column space of $\,X$, and $\,Q \in G(2,q)$ is recovered as the row space of $\,X$.

Figures (3)

  • Figure 1: Network representation of the matrix $(m_{i,j})$ for $k=4$ and $n=9$. The polynomial $m_{i,j}$ is a sum over all north-east paths from node $i$ to node $4+j$. There are $35$ such paths for $i=4,j=5$.
  • Figure 2: The tropical variety defined by the $3\times3$ determinant. Image taken from Berndbook.
  • Figure :

Theorems & Definitions (14)

  • Example 2.1: $k=2$
  • Example 2.2: $k=3$
  • Example 3.1: $k=2$
  • Example 3.2: $k=n-2$
  • Example 3.3: $k=3,n=6$
  • Example 3.4: $k=4,n=9$
  • Example 5.1: $k=2$
  • Example 5.2: $k=n-2$
  • Example 5.3: $k=3,n=6$
  • Example 6.1
  • ...and 4 more