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Orbital Structure of 6-Point MHV Gravity Forms

Zachary G. Craig

TL;DR

This work investigates the 6-point MHV gravity amplitude through a logarithmic $3$-form on the De Concini--Procesi compactification, constrained to physical poles and factorization. The authors use the Orlik--Solomon algebra on the channel arrangement with $S_3\times S_3$ symmetry to search for a global $d\log$ form, finding that a single line of compatible-boundary candidates is obstructed by crossing constraints. They resolve this with an orbit-mixed form spanning $20$ permutation images, implying the amplitude lives in a rank-$20$ local system whose monodromy mixes chamber representations. The results support an arithmetic-geometry perspective on scattering amplitudes, proposing a CHY-Galois correspondence and an Arithmetic Amplituhedron conjecture, and they provide reproducible code and data.

Abstract

We search for a logarithmic 3-form representing the 6-point MHV gravity amplitude, requiring poles only on physical channels and residues matching factorization. Working in the Orlik-Solomon algebra on a De Concini-Procesi wonderful model, we restrict to the S3 x S3 invariant subspace and impose factorization boundary-by-boundary. The intersection of ten compatible 3|3 channels with two compatible 2-particle channels collapses to a unique one-dimensional candidate line. However, enforcing factorization on a crossing channel obstructs any single-valued global representative. We find that an orbit-mixed form, a linear combination over 20 permutation images indexed by bipartitions, satisfies the crossing constraint. This is consistent with a local-system picture: the global gravity form lives in a rank-20 bundle whose monodromy mixes bipartition chambers rather than flipping a single sign. All code and data are provided for reproducibility.

Orbital Structure of 6-Point MHV Gravity Forms

TL;DR

This work investigates the 6-point MHV gravity amplitude through a logarithmic -form on the De Concini--Procesi compactification, constrained to physical poles and factorization. The authors use the Orlik--Solomon algebra on the channel arrangement with symmetry to search for a global form, finding that a single line of compatible-boundary candidates is obstructed by crossing constraints. They resolve this with an orbit-mixed form spanning permutation images, implying the amplitude lives in a rank- local system whose monodromy mixes chamber representations. The results support an arithmetic-geometry perspective on scattering amplitudes, proposing a CHY-Galois correspondence and an Arithmetic Amplituhedron conjecture, and they provide reproducible code and data.

Abstract

We search for a logarithmic 3-form representing the 6-point MHV gravity amplitude, requiring poles only on physical channels and residues matching factorization. Working in the Orlik-Solomon algebra on a De Concini-Procesi wonderful model, we restrict to the S3 x S3 invariant subspace and impose factorization boundary-by-boundary. The intersection of ten compatible 3|3 channels with two compatible 2-particle channels collapses to a unique one-dimensional candidate line. However, enforcing factorization on a crossing channel obstructs any single-valued global representative. We find that an orbit-mixed form, a linear combination over 20 permutation images indexed by bipartitions, satisfies the crossing constraint. This is consistent with a local-system picture: the global gravity form lives in a rank-20 bundle whose monodromy mixes bipartition chambers rather than flipping a single sign. All code and data are provided for reproducibility.
Paper Structure (7 sections, 1 theorem, 6 equations)

This paper contains 7 sections, 1 theorem, 6 equations.

Table of Contents

  1. Motivation
  2. Setup and constraints
  3. Compatible boundary intersection
  4. Crossing boundary obstruction and orbit mixing
  5. Reproducibility
  6. Discussion
  7. Future Directions: The Arithmetic Amplituhedron Conjecture

Key Result

Proposition 4.3

Let $v$ span the one-dimensional intersection $V_{\mathrm{comp}}$. The crossing-boundary factorization constraint at $(1,4)$ fails for $v$ and for the bipartite $\mathbb{Z}_2$ sign-twisted vector. However, there exists an orbit-mixed form in the span of $\{v_i\}$ that satisfies the crossing constrai

Theorems & Definitions (6)

  • Definition 4.1: Bipartite $\mathbb{Z}_2$ sign twist
  • Definition 4.2: Orbit-mixed form
  • Proposition 4.3: Crossing test (computational)
  • Remark 4.4
  • Remark 6.1: Bipartition Structure
  • Conjecture 7.1: Chamber-Galois Correspondence