Orientations of BCFW Charts on the Grassmannian

TL;DR

This paper develops an algorithm to determine orientations of BCFW charts on the Grassmannian, enabling precise sign calculations that ensure correct residue cancellations in scattering amplitude computations.

Contribution

It introduces a simple method to compute relative orientations between Grassmannian charts, generalizing to sequences of transpositions and ensuring accurate sign assignments.

Findings

Signed boundary operator that squares to zero

All non-local poles cancel exactly with correct signs

Algorithm applicable to sequences of transpositions

Abstract

The Grassmannian formulation of $\mathcal{N}=4$ super Yang-Mills theory expresses tree-level scattering amplitudes as linear combinations of residues from certain contour integrals. BCFW bridge decompositions using adjacent transpositions simplify the evaluation of individual residues, but orientation information is lost in the process. We present a straightforward algorithm to compute relative orientations between the resulting coordinate charts, and we show how to generalize the technique for charts corresponding to sequences of any not-necessarily-adjacent transpositions. As applications of these results, we demonstrate the existence of a signed boundary operator that manifestly squares to zero and prove via our algorithm that any residues appearing in the tree amplitude sum are decorated with appropriate signs so all non-local poles cancel exactly, not just mod 2 as in previous works.