How gluon leading singularities discover curves on surfaces

TL;DR

This paper reveals how leading singularities in gluon amplitudes can be understood through a combinatorial problem involving covering graphs with non-overlapping curves on surfaces, connecting surface topology with amplitude calculations.

Contribution

It introduces a novel combinatorial approach to determine leading singularities in gluon amplitudes using surface topology and graph coverings, applicable at any loop order.

Findings

Leading singularities correspond to non-overlapping curve coverings of graphs.

The combinatorial problem matches the surfaceology formulation of gluons.

The approach accounts for spin sums and ghost contributions at loop level.

Abstract

We study the leading singularities for pure gluon amplitudes obtained by on-shell gluing of three-particle amplitudes for an arbitrary graph in any number of dimensions. By encoding the polarization vector contractions in a graphical way, on-shell gluing "discovers" curves on surfaces, and we find that the leading singularity is determined by a simple combinatorial question: what are all ways of covering the graph with non-overlapping curves such that each edge is covered exactly once? This precisely matches the formula from the surfaceology formulation of gluons, where the leading singularities are given by maximal residues, with the combinatorial problem arising from the linearized form of the $u$ variables. At loop-level we describe how the novelties associated with spin sums (related with the need for ghosts when working off-shell using Lagrangians) can be easily encoded in this combinatorial picture. Matching the leading singularities also lets us settle an open question in the surface formulation of gluons, determining the exponents of the closed curves at any loop order.