Singular Points on Positroid Varieties and Planar N=4 Supersymmetric Yang-Mills Theory

Abstract

Positroid varieties $\Pi_f$ provide a decomposition of the Grassmannian $Gr(k,n)$; they can be enumerated using bounded affine permutations ($f$) which have bijections with a number of interesting combinatorial objects. Furthermore, (the nonnegative part of) positroid varieties parameterize the space that is integrated over when calculating amplitudes in planar N=4 supersymmetric Yang-Mills theory. The main question we answer in this thesis is whether a positroid variety $\Pi_f$ has any geometric singularities. We show that it is sufficient to check singularity at the $T$-fixed points ($\lambda$) and we can obtain the multiplicity at these points $\lambda$ by calculating the equivariant cohomology of $\Pi_f$ restricted to the point. We give 2 ways of doing this: (1) A diagrammatic way using affine pipe dreams and (2) A computational method. For (2), we have written code that does the computation and outputs the multiplicity (files at josephflueg.github.io). We have included tables listing the multiplicities of all the points on positroid varieties up to $n=6$. We also describe an ordering on pairs $(\Pi_f,\lambda)$ given by deletion/contraction that interacts nicely with smoothness, and describe the relationship between affine pipe dreams and finite pipe dreams. In Part II of this thesis connects with physics of planar N=4 SYM. We briefly introduce quantum field theory, leading singularities, and positroids in N=4 SYM. We then explain Britto-Cachazo-Feng-Witten (BCFW) recursion and the BCFW bridge decomposition of an on-shell diagram. We describe how to build an on-shell diagram on a pipe dream using BCFW bridges. Finally, we work out some combinatorics related to inverse soft factors for on-shell diagrams and explore whether singularities in positroid varieties have relevance to amplitudes. This is a revised version of my thesis originally written in August 2024.