Unraveling L_{n,k}: Grassmannian Kinematics

TL;DR

This paper develops a method to identify and analyze leading singularities in N=4 super Yang-Mills theory using Grassmannian integrals, revealing new structures at tree, one-loop, and higher loops.

Contribution

It provides a systematic way to identify residues corresponding to leading singularities in Grassmannian integrals, including explicit calculations at various loop levels and new kinematic structures.

Findings

Explicit identification of all tree and one-loop leading singularities.

Analysis of two-loop and four-loop leading singularities.

Discovery of a new kinematic structure involving double square roots.

Abstract

It was recently proposed that the leading singularities of the S-Matrix of N = 4 super Yang-Mills theory arise as the residues of a contour integral over a Grassmannian manifold, with space-time locality encoded through residue theorems generalizing Cauchy's theorem to more than one variable. We provide a method to identify the residue corresponding to any leading singularity, and we carry this out very explicitly for all leading singularities at tree level and one-loop. We also give several examples at higher loops, including all generic two-loop leading singularities and an interesting four-loop object. As a special case we consider a 12-pt N^4MHV leading singularity at two loops that has a new kinematic structure involving double square roots. Our analysis results in a simple picture for how the topological structure of loop graphs is reflected in various substructures within the Grassmannian.