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

@article{Agarwala2018WilsonLI, title={Wilson loops in SYM \$N=4\$ do not parametrize an orientable space}, author={Susama Agarwala and Cameron Marcott}, journal={arXiv: Mathematical Physics}, year={2018} }

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 the Grassmannian, $\mathcal{W}_{k,cn}$. Furthermore, we explicitly show that this bundle is non-orientable in the majority of the cases, and conjecture that it is non-orientable in the remaining situation. Using the combinatorics of the Deodhar decomposition of the Grassmannian, we identify subspaces…

## 4 Citations

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

- Mathematics
- 2021

This paper shows that not only do the codimension one spurious poles of NMHV tree level diagrams in N=4 SYM theory cancel in the tree level amplitude as expected, but their vanishing loci have a…

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

- MathematicsCanadian Journal of Mathematics
- 2021

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…

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

- MathematicsCanadian Journal of Mathematics
- 2021

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…

### The twistor Wilson loop and the amplituhedron

- PhysicsJournal of High Energy Physics
- 2018

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.…

## References

SHOWING 1-10 OF 27 REFERENCES

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

- MathematicsAnnales de l’Institut Henri Poincaré D
- 2022

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…

### Wilson Loop Diagrams and Positroids

- Mathematics
- 2017

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…

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

- Physics
- 2010

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…

### Parametrizations of flag varieties

- Mathematics
- 2003

For the flag variety G/B of a reductive algebraic group G we define and describe explicitly a certain (set-theoretical) cross-section φ : G/B → G. The definition of φ depends only on a choice of…

### Positroid varieties: juggling and geometry

- MathematicsCompositio Mathematica
- 2013

Abstract While the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, it turns out that the intersection of only the cyclic shifts of one Bruhat…

### Total positivity for cominuscule Grassmannians

- Mathematics
- 2007

In this paper we explore the combinatorics of the nonneg- ative part (G/P )≥0 of a cominuscule Grassmannian. For each such Grassmannian we define Γ — certain fillings of generalized Young diagrams…

### The correlahedron

- Geology
- 2017

A bstractWe introduce a new geometric object, the correlahedron, which we conjecture to be equivalent to stress-energy correlators in planar N=4$$ \mathcal{N}=4 $$ super Yang-Mills. Re-expressing the…

### Decompositions of amplituhedra

- MathematicsAnnales de l’Institut Henri Poincaré D
- 2020

The (tree) amplituhedron A(n,k,m) is the image in the Grassmannian Gr(k,k+m) of the totally nonnegative part of Gr(k,n), under a (map induced by a) linear map which is totally positive. It was…

### The twistor Wilson loop and the amplituhedron

- PhysicsJournal of High Energy Physics
- 2018

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.…