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 M HV 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, M V , 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.

The Amplituhedron BCFW Triangulation

“…Remark 4.7. This functionary and similar ones have appeared in the literature before, for example in [Lam15] and in [AM21].…”

“…Our basic ingredient in separating cells is the twistor coordinates, similarly to Parisi, Sherman-Bennett and Williams [PSW21]. Our analysis of boundaries is reminiscent with that of Agarwala and Marcott [AM21]. We use the domino form by Karp, Williams, Zhang, and Thomas [KWZ20] to represent BCFW cells.…”

“…These are given by particular plabic graphs in [ABC `16] and by pairs of noncrossing lattice walks in [KWZ20], who also described additional equivalent viewpoints using binary trees, Dyck paths and domino matrices. For our purposes, it is convenient to start from yet another representation, which is a variant of the Wilson loop diagrams used by Agarwala, Marin-Amat and Marcott [AM17,AM21]. We start with a definition of these combinatorial structures.…”

“…Remark 2.2. Agarwala et al [AM17,AM21] use the term propagators for chords, and label the markers by Z's rows: Z 1 , . .…”

“…Remark 7.16. In [AM21] an interesting analysis of boundaries and their cancellation was performed for another related physical model. In that model some other class of chord diagrams were used for indexing cells.…”