Wilson loops in SYM $\mathcal{N}=4$ do not parametrize an orientable space

Agarwala, Susama; Marcott, Cameron

doi:10.4171/aihpd/111

Cited by 3 publications

(5 citation statements)

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“…Here, we show that for k D 2 and n D 6, the Wilson loop diagrams define a 6-dimensional subspace of G R;0 .k; n/, or conjecturally, a 3k-dimensional subspace of the positive Grassmannian. Subsequent work has verified this conjecture [4,23], and shown that if one includes the gauge vector that is necessary in the N k MHV calculations (but not in the BCFW computations), one gets a 4k-dimensional subspaces of the full Grassmannian G R .k; n C 1/, see [6].…”

Section: Introductionmentioning

confidence: 90%

A study in $\mathbb{G}_{\mathbb{R}, \geq 0}(2,6)$: from the geometric case book of Wilson loop diagrams and SYM $N=4$

Agarwala

Fryer

2022

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

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We study the geometry underlying the Wilson loop diagram approach to calculating scattering amplitudes in Supersymmetric Yang Mills (SYM) N D 4. In particular, we study the smallest non-trivial multi-propagator case, consisting of 2 propagators on 6 vertices. We do this by translating the integrals of the theory to the combinatorics of the positive geometry each diagram represents, specifically identifying the positroid cells defined by each diagram and the homology of the subcomplex they collectively generate in G R;0 .2; 6/. We verify the conjecture that the spurious singularities of the volume functional do all cancel on the codimension 1 boundaries of these cells, in this case. We also show that how the spurious singularities cancel is actually much more complicated than previously understood. The direct calculation laid out in this paper identifies many intricacies and artifacts of the geometry of Wilson loop diagram that need further study.

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Section: Introductionmentioning

confidence: 90%

A study in $\mathbb{G}_{\mathbb{R}, \geq 0}(2,6)$: from the geometric case book of Wilson loop diagrams and SYM $N=4$

Agarwala

Fryer

2022

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

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“…Let q be the propagator indicated by the dashed line and p the propagator indicated by a dotted line. Then q ∈ Q F , and F supports two ends of p. However, all the ends of q not supported by vertices in F are in one cyclic interval in the complement of F , namely, [7,12]. Therefore, by Lemma 5.19, M(W ) is a positroid, even though W has crossing propagators.…”

Section: Positive Generalized Wilson Loop Diagramsmentioning

confidence: 99%

“…Moving to boundaries of admissible ordinary Wilson loop diagrams with |P| > 2, we do not yet have sufficient moves to obtain all boundaries at all codimensions. The moves are sufficient to generate the codimension 1 boundaries of all diagrams in WLD (3,6), WLD (3,7), and the majority of diagrams in WLD (3,8). The missing boundaries from WLD (3,8) then are particularly interesting.…”

Section: Graphical Calculus For Boundaries Of Wilson Loop Diagrams Mo...mentioning

confidence: 99%

“…For example, for the diagram in Example 2.4, we may set S = [9] and S 1 = [8,2] and S 2 = [3,7]. In this case, the sets S 1 and S 2 form a partition of [n] and the set End(S 1 ) and End(S 2 ) form a partition of E(P).…”

Section: The X-merge and Split Movementioning

confidence: 99%

“…Wilson loop diagrams are a variant of the notion of chord diagram that was introduced in high energy physics in order to compute scattering amplitudes in N = 4 super Yang Mills theory (SYM). Each admissible Wilson loop diagram is associated to a positroid, which has led to their study from a combinatorial [5,6], geometric [7,8,3], and matroidal [2,3] point of view, in addition to their investigation from a physics perspective [2,8,16,20].…”

Section: Introductionmentioning

confidence: 99%

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

Agarwala

Fryer²,

Yeats³

2021

Can. J. Math.-J. Can. Math.

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Wilson loop diagrams are an important tool in studying scattering amplitudes of SYM = 4 theory and are known by previous work to be associated to positroids. We characterize the conditions under which two Wilson loop diagrams give the same positroid, prove that an important subclass of subdiagrams (exact subdiagrams) corresponds to uniform matroids, and enumerate the number of different Wilson loop diagrams that correspond to each positroid cell. We also give a correspondence between those positroids which can arise from Wilson loop diagrams and directions in associahedra.

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

Agarwala¹,

Marcott²

2023

Advances in Applied Mathematics

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Wilson loops in SYM $\mathcal{N}=4$ do not parametrize an orientable space

Cited by 3 publications

References 0 publications

A study in $\mathbb{G}_{\mathbb{R}, \geq 0}(2,6)$: from the geometric case book of Wilson loop diagrams and SYM $N=4$

A study in $\mathbb{G}_{\mathbb{R}, \geq 0}(2,6)$: from the geometric case book of Wilson loop diagrams and SYM $N=4$

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

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

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