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 NkMHV 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, Σ, and a minimal matrix representation of it in terms of independent variable valued matrices, MV, one can define a polynomial, R(V) that is uniquely defined by the Grassmann necklace, I, of the positroid cell. The vanishing locus of R(V) lies on the boundary of the positive variety Σ‾∖Σ, but not all boundaries intersect the vanishing loci of a factor of R(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.