Orbital Structure of 6-Point MHV Gravity Forms

TL;DR

This paper investigates the mathematical structure of 6-point MHV gravity amplitudes, finding a specific form that satisfies physical and factorization constraints using advanced algebraic geometry, revealing a complex monodromy structure.

Contribution

It introduces a novel orbit-mixed form that satisfies crossing and factorization constraints, advancing the understanding of gravity amplitude representations.

Findings

A unique candidate line for the 6-point MHV gravity amplitude was identified.

An orbit-mixed form satisfying crossing constraints was constructed.

The global gravity form resides in a rank-20 bundle with complex monodromy.

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.