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Moduli spaces in positive geometry

Thomas Lam

Abstract

These are lecture notes for five lectures given at MPI Leipzig in May 2024. We study the moduli space M_{0,n} of n distinct points on P^1 as a positive geometry and a binary geometry. We develop mathematical formalism to study Cachazo-He-Yuan's scattering equations and the associated scalar and Yang-Mills amplitudes. We discuss open superstring amplitudes and relations to tropical geometry.

Moduli spaces in positive geometry

Abstract

These are lecture notes for five lectures given at MPI Leipzig in May 2024. We study the moduli space M_{0,n} of n distinct points on P^1 as a positive geometry and a binary geometry. We develop mathematical formalism to study Cachazo-He-Yuan's scattering equations and the associated scalar and Yang-Mills amplitudes. We discuss open superstring amplitudes and relations to tropical geometry.
Paper Structure (58 sections, 55 theorems, 224 equations)

This paper contains 58 sections, 55 theorems, 224 equations.

Table of Contents

  1. Positive geometry of ${\mathcal{M}}_{0,n}$
  2. Matrices, Moduli space, and the Grassmannian
  3. Cross ratios
  4. Dihedral coordinates
  5. Affine closure and compactification
  6. Positive coordinates
  7. Canonical form on ${\mathcal{M}}_{0,n}$
  8. Positive geometry
  9. Exercises and Problems
  10. Canonical form of ${\rm Gr}(2,n)_{\geq 0}$
  11. Canonical class of $\overline{{\mathcal{M}}}_{0,n}$ and zeroes of $\Omega_{0,n}$
  12. Binary geometry
  13. Binary geometries and $u$-equations
  14. One-dimensional binary geometries
  15. Two-dimensional binary geometries
  16. ...and 43 more sections

Key Result

Lemma 1.2

The action of ${\rm PGL}(2) \times T \cong {\rm GL}(2) \times T'$ on ${\rm Mat}_{2,n}^\circ$ is free.

Theorems & Definitions (156)

  • Definition 1.1
  • Lemma 1.2
  • Proposition 1.3
  • Definition 1.4
  • Lemma 1.5
  • Proposition 1.6
  • proof
  • Theorem 1.7: BroAHLcluster
  • Proposition 1.8
  • proof
  • ...and 146 more