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Minimal Kinematics on $\mathcal{M}_{0,n}$

Nick Early, Anaëlle Pfister, Bernd Sturmfels

Abstract

Minimal kinematics identifies likelihood degenerations where the critical points are given by rational formulas. These rest on the Horn uniformization of Kapranov-Huh. We characterize all choices of minimal kinematics on the moduli space $\mathcal{M}_{0,n}$. These choices are motivated by the CHY model in physics and they are represented combinatorially by 2-trees. We compute 2-tree amplitudes, and we explore extensions to non-planar on-shell diagrams, here identified with the hypertrees of Castravet-Tevelev.

Minimal Kinematics on $\mathcal{M}_{0,n}$

Abstract

Minimal kinematics identifies likelihood degenerations where the critical points are given by rational formulas. These rest on the Horn uniformization of Kapranov-Huh. We characterize all choices of minimal kinematics on the moduli space . These choices are motivated by the CHY model in physics and they are represented combinatorially by 2-trees. We compute 2-tree amplitudes, and we explore extensions to non-planar on-shell diagrams, here identified with the hypertrees of Castravet-Tevelev.
Paper Structure (5 sections, 9 theorems, 67 equations, 1 figure)

This paper contains 5 sections, 9 theorems, 67 equations, 1 figure.

Table of Contents

  1. Introduction
  2. 2-trees
  3. Horn matrices
  4. Amplitudes
  5. On-Shell Diagrams and Hypertrees

Key Result

Proposition 1.1

For a general choice of $s_{ij}$, the scattering potential $L$ has $(n-3)!$ complex critical points on $\mathcal{M}_{0,n}$. If the $s_{ij}$ are real numbers, then all $(n-3)!$ critical points are real.

Figures (1)

  • Figure 1: Combinatorics of the octahedral hypertree amplitude $m_T$.

Theorems & Definitions (28)

  • Proposition 1.1
  • Example 1.2: $n=4$
  • Example 1.3: $n=6$
  • Theorem 1.4
  • Example 1.5: $n=6$
  • Lemma 2.1
  • proof
  • Example 2.2: $n=6$
  • proof : Proof of Theorem \ref{['thm:vier']}
  • Theorem 3.1
  • ...and 18 more