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Rado matroids and a graphical calculus for boundaries of Wilson loop diagrams

Susama Agarwala, Colleen Delaney, Karen Yeats

Abstract

We study the boundaries of the positroid cells which arise from N = 4 super Yang Mills theory. Our main tool is a new diagrammatic object which generalizes the Wilson loop diagrams used to represent interactions in the theory. We prove conditions under which these new generalized Wilson loop diagrams correspond to positroids and give an explicit algorithm to calculate the Grassmann necklace of said positroids. Then we develop a graphical calculus operating directly on noncrossing generalized Wilson loop diagrams. In this paradigm, applying diagrammatic moves to a generalized Wilson loop diagram results in new diagrams that represent boundaries of its associated positroid, without passing through cryptomorphisms. We provide a Python implementation of the graphical calculus and use it to show that the boundaries of positroids associated to ordinary Wilson loop diagram are generated by our diagrammatic moves in certain cases.

Rado matroids and a graphical calculus for boundaries of Wilson loop diagrams

Abstract

We study the boundaries of the positroid cells which arise from N = 4 super Yang Mills theory. Our main tool is a new diagrammatic object which generalizes the Wilson loop diagrams used to represent interactions in the theory. We prove conditions under which these new generalized Wilson loop diagrams correspond to positroids and give an explicit algorithm to calculate the Grassmann necklace of said positroids. Then we develop a graphical calculus operating directly on noncrossing generalized Wilson loop diagrams. In this paradigm, applying diagrammatic moves to a generalized Wilson loop diagram results in new diagrams that represent boundaries of its associated positroid, without passing through cryptomorphisms. We provide a Python implementation of the graphical calculus and use it to show that the boundaries of positroids associated to ordinary Wilson loop diagram are generated by our diagrammatic moves in certain cases.
Paper Structure (36 sections, 33 theorems, 69 equations, 2 figures, 2 algorithms)

This paper contains 36 sections, 33 theorems, 69 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Generalized Wilson loop diagrams
  3. General set up
  4. Generalized Wilson loop diagrams with non-crossing propagators
  5. Background
  6. Physics motivation
  7. Matroidal background
  8. Matroids
  9. Transversal and Rado matroids
  10. Positroids
  11. Matroids of generalized Wilson loop diagrams
  12. Generalized Wilson loop diagrams and their matroids
  13. Properties of matroids associated to generalized Wilson loop diagrams
  14. Minimal and capacity-ranked generalized Wilson loop diagrams
  15. Generalized Wilson loop diagrams representing the same matroid
  16. ...and 21 more sections

Key Result

Lemma 2.13

Let $W = (\mathcal{P}, [n])$ be a non-crossing generalized Wilson loop diagram, and $a \in C$. Take $p,q\in \textrm{Prop}(E_a)$. Then either for all $b \in \mathcal{E}(p) \setminus E_a$ and all $c \in \mathcal{E}(q) \setminus E_a$ we get $b\leq_a c$, or the same holds with $p$ and $q$ reversed.

Figures (2)

  • Figure 1: As described in Example \ref{['eg: half props clear']}, if a single-ended propagator has support $V(p) = \{i, i+1\}$, one may change the support of any ends sharing the support to either be $i$ or $i+1$ without changing the associated Rado matroid. If the single-ended propagator has support $i$, then one may remove $i$ from any end with $i$ in its support without changing the associated Rado matroid.
  • Figure 2: Enumerating the Le diagrams corresponding to the boundaries of the positroid associated to a Wilson loop diagram requires passing through several intermediate combinatorial objects.

Theorems & Definitions (139)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6
  • Example 2.7
  • Definition 2.8
  • Example 2.9
  • Remark 2.10
  • ...and 129 more