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

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.