Cancellation of spurious poles in N=4 SYM: physical and geometric

TL;DR

This paper demonstrates that spurious poles in N=4 SYM tree amplitudes cancel out and reveals their geometric interpretation within the positive Grassmannian framework, linking algebraic and geometric structures.

Contribution

It provides a geometric understanding of spurious pole cancellation in N=4 SYM using positroid varieties and Grassmannian representations, extending the theoretical insight into amplitude singularities.

Findings

Spurious poles cancel in N=4 SYM tree amplitudes.

Vanishing loci of certain polynomials relate to boundaries of positive Grassmannian.

Geometric interpretation connects algebraic pole cancellation to Grassmannian boundaries.

Abstract

This paper shows that not only do the codimension one spurious poles of $N^k MHV$ tree level diagrams in N=4 SYM theory cancel in the tree level amplitude as expected, but their vanishing loci have a geometric interpretation that is tightly connected to their representation in the positive Grassmannians. In general, given a positroid variety, $\Sigma$, and a minimal matrix representation of it in terms of independent variable valued matrices, $M_{\mathcal{V}}$, one can define a polynomial, $R({\mathcal{V}})$ that is uniquely defined by the Grassmann necklace, ${\mathcal{I}}$, of the positroid cell. The vanishing locus of $R({\mathcal{V}})$ lies on the boundary of the positive variety $\overline{\Sigma} \setminus \Sigma$, but not all boundaries intersect the vanishing loci of a factor of $R({\mathcal{V}})$. We use this to show that the codimension one spurious poles of N=4 SYM, represented in twistor space, cancel in the tree level amplitude.