Antipodal self-duality of square fishnet graphs

TL;DR

This paper demonstrates that square fishnet graphs in a specific quantum field theory exhibit an antipodal self-duality, revealing a deep symmetry involving kinematic and algebraic transformations of the associated Feynman integrals.

Contribution

It introduces the concept of antipodal self-duality in square fishnet graphs, connecting geometric and algebraic symmetries in these Feynman integrals.

Findings

Square fishnet graphs are invariant under combined kinematic and antipode maps.

The antipodal self-duality holds for all integer sizes of the grid.

This symmetry links the structure of Feynman integrals to Hopf algebra properties.

Abstract

In strongly-deformed planar ${\cal N}=4$ super-Yang-Mills theory, or fishnet theory, a point-split single-trace correlation function of four dimension-$m$ scalar operators is given by a single Feynman integral, which involves integrating over locations of a $m\times m$ grid of points. We show that for any integer $m$ this square fishnet graph is invariant under the combined action of a kinematic map and the antipode map of the Hopf algebra on multiple polylogarithms, i.e. it possesses an antipodal self-duality.