Antipodal Symmetry of Two-Loop MHV Amplitudes

TL;DR

This paper conjectures that all two-loop MHV amplitudes in planar N=4 super-Yang-Mills theory exhibit an antipodal symmetry acting on symbol letters, with explicit maps for up to eight particles and evidence for all multiplicities.

Contribution

It introduces a conjecture that these amplitudes have a specific antipodal symmetry, providing explicit symbol-level maps and evidence across all particle multiplicities.

Findings

Explicit symmetry map for up to eight particles.

Evidence supporting symmetry at all multiplicities.

The symmetry acts as an isomorphism on Newton polytope fans.

Abstract

I present a conjecture that all two-loop MHV amplitudes in planar $\mathcal{N} = 4$ super-Yang-Mills theory possess an antipodal symmetry when evaluated on parity-even kinematics. The symmetry acts as a change of basis on the symbol letters, followed by the antipode operation associated with the Hopf algebra structure of multiple polylogarithms. At the symbol level, I provide the symmetry map explicitly for amplitudes with up to eight external particles, and also provide evidence at all multiplicities. Intriguingly, the map acts as an isomorphism on the normal fans of the Newton polytopes of the symbol letters. The conjectured symmetry is one of the rare known cases where the antipode map shows up in physically important examples.