# Wilson Loop Diagrams and Positroids

@article{Agarwala2015WilsonLD, title={Wilson Loop Diagrams and Positroids}, author={Susama Agarwala and Eloi Marin-Amat}, journal={Communications in Mathematical Physics}, year={2015}, volume={350}, pages={569-601} }

In this paper, we study a new application of the positive Grassmannian to Wilson loop diagrams (or MHV diagrams) for scattering amplitudes in N= 4 Super Yang–Mill theory (N = 4 SYM). There has been much interest in studying this theory via the positive Grassmannians using BCFW recursion. This is the first attempt to study MHV diagrams for planar Wilson loop calculations (or planar amplitudes) in terms of positive Grassmannians. We codify Wilson loop diagrams completely in terms of matroids…

## 14 Citations

### Combinatorics of the geometry of Wilson loop diagrams II: Grassmann necklaces, dimensions, and denominators

- 2021

Mathematics

Canadian Journal of Mathematics

Abstract Wilson loop diagrams are an important tool in studying scattering amplitudes of SYM
$N=4$
theory and are known by previous work to be associated to positroids. In this paper, we study the…

### A study in 𝔾ℝ,≥0: from the geometric case book of Wilson loop diagrams and SYM N=4

- 2022

Mathematics

Annales de l’Institut Henri Poincaré D

We study the geometry underlying the Wilson loop diagram approach to calculating scattering amplitudes in the gauge theory of Supersymmetric Yang Mills (SYM) N=4. By applying the tools developed to…

### Combinatorics of the geometry of Wilson loop diagrams I: equivalence classes via matroids and polytopes

- 2021

Mathematics

Canadian Journal of Mathematics

Abstract Wilson loop diagrams are an important tool in studying scattering amplitudes of SYM
$N=4$
theory and are known by previous work to be associated to positroids. We characterize the…

### Wilson loops in SYM $N=4$ do not parametrize an orientable space

- 2018

Mathematics

In this paper we explore the geometric space parametrized by (tree level) Wilson loops in SYM $N=4$. We show that, this space can be seen as a vector bundle over a totally non-negative subspace of…

### Total positivity is a quantum phenomenon: the grassmannian case

- 2019

Mathematics

The main aim of this paper is to establish a deep link between the totally nonnegative grassmannian and the quantum grassmannian. More precisely, under the assumption that the deformation parameter…

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

- 2023

Mathematics

Advances in Applied Mathematics

### The twistor Wilson loop and the amplituhedron

- 2018

Physics

Journal of High Energy Physics

A bstractThe amplituhedron provides a beautiful description of perturbative superamplitude integrands in N=4$$ \mathcal{N}=4 $$ SYM in terms of purely geometric objects, generalisations of polytopes.…

### Combinatorics of Grassmannian Decompositions

- 2019

Mathematics

This thesis studies several combinatorially defined families of subsets of the Grassmannian of k-dimensional subspaces of R, Gr(k, n). We introduce and study a family of subsets called “basis shape…

### Basis shape loci and the positive Grassmannian

- 2019

Mathematics

A basis shape locus takes as input data a zero/nonzero pattern in an $n \times k$ matrix, which is equivalent to a presentation of a transversal matroid. The locus is defined as the set of points in…

### 3 0 A pr 2 01 9 Basis shape loci and the positive Grassmannian

- 2019

Mathematics

A basis shape locus takes as input data a zero/nonzero pattern in an n× k matrix, which is equivalent to a presentation of a transversal matroid. The locus is defined as the set of points in Gr(k, n)…

## 21 References

### Scattering Amplitudes and the Positive Grassmannian

- 2012

Mathematics

We establish a direct connection between scattering amplitudes in planar four-dimensional theories and a remarkable mathematical structure known as the positive Grassmannian. The central physical…

### Scattering amplitudes and Wilson loops in twistor space

- 2011

Mathematics

This paper reviews the recent progress in twistor approaches to Wilson loops, amplitudes and their duality for super-Yang–Mills. Wilson loops and amplitudes are derived from first principles using…

### The complete planar S-matrix of $ \mathcal{N} = 4 $ SYM as a Wilson loop in twistor space

- 2010

Physics

We show that the complete planar S-matrix of $ \mathcal{N} = 4 $ super Yang-Mills — including all NkMHV partial amplitudes to all loops — is equivalent to the correlation function of a supersymmetric…

### MHV diagrams in twistor space and the twistor action

- 2011

Physics

MHV diagrams give an efficient Feynman diagram-like formalism for calculating gauge theory scattering amplitudes on momentum space. Although they arise as the Feynman diagrams from an action on…

### Maximally helicity-violating diagrams in twistor space and the twistor action

- 2012

Physics

MHV diagrams give an efficient Feynman diagram-like formalism for calculating gauge theory scattering amplitudes on momentum space. Although they arise as the Feynman diagrams from an action on…

### Grassmannians for scattering amplitudes in 4d N=4$$ \mathcal{N}=4 $$ SYM and 3d ABJM

- 2014

Mathematics

A bstractScattering amplitudes in 4d N=4$$ \mathcal{N}=4 $$ super Yang-Mills theory (SYM) can be described by Grassmannian contour integrals whose form depends on whether the external data is encoded…

### The Amplituhedron

- 2013

Mathematics

A bstractPerturbative scattering amplitudes in gauge theories have remarkable simplicity and hidden infinite dimensional symmetries that are completely obscured in the conventional formulation of…

### MHV vertices and tree amplitudes in gauge theory

- 2004

Physics

As an alternative to the usual Feynman graphs, tree amplitudes in Yang-Mills theory can be constructed from tree graphs in which the vertices are tree level MHV scattering amplitudes, continued off…

### Matching polytopes, toric geometry, and the totally non-negative Grassmannian

- 2009

Mathematics

In this paper we use toric geometry to investigate the topology of the totally non-negative part of the Grassmannian, denoted (Grk,n)≥0. This is a cell complex whose cells ΔG can be parameterized in…