[Paper Review] Wilson loops in SYM $N=4$ do not parametrize an orientable space

Susama Agarwala, Cameron Marcott|arXiv (Cornell University)|Jul 14, 2018

Advanced Combinatorial Mathematics6 references3 citations

TL;DR

This paper demonstrates that the geometric space parametrized by tree-level Wilson loops in $\mathcal{N}=4$ SYM theory is non-orientable for most cases, arising from a vector bundle over a positroid cell in the real Grassmannian $\mathrm{Gr}_{\mathbb{R}}(k,n+1)$ that fails to admit a global volume form. The non-orientability stems from transition functions with negative determinants in the fiber bundle structure, challenging the naive geometric interpretation of scattering amplitudes via volume forms.

ABSTRACT

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 $Σ(W) \subset \mathcal{W}_{k,n}$ for which the restricted bundle lies outside the positive Grassmannian. Finally, while probing the combinatorics of the Deodhar decomposition, we give a diagrammatic algorithm for reading equations determining each Deodhar component as a semialgebraic set.

Motivation & Objective

• To investigate the geometric structure underlying tree-level Wilson loop diagrams in $\mathcal{N}=4$ SYM theory.
• To determine whether the space parametrized by these diagrams admits a global volume form, which is essential for geometric amplitude computations.
• To analyze the orientability of the $4k$-dimensional fiber bundle over positroid cells in the Grassmannian $\mathrm{Gr}_{\mathbb{R}}(k,n+1)$ defined by Wilson loop diagrams.
• To identify conditions under which the bundle fails to be orientable, particularly through transition functions with negative determinants.
• To explore the implications of non-orientability for the geometric program of understanding scattering amplitudes via the Amplituhedron and related volumes.

Proposed method

• The authors use the Deodhar decomposition of the real Grassmannian $\mathrm{Gr}_{\mathbb{R}}(k,n)$ to refine the positroid stratification and analyze finer geometric structures beyond the positive part.
• They model the Wilson loop parametrization as a fiber bundle $\pi^{-1}(\Sigma(W))$ over a positroid cell $\Sigma(W)$ in $\mathrm{Gr}_{\mathbb{R}}(k,n)$, with fibers in $\mathrm{Gr}_{\mathbb{R}}(k,n+1)$.
• Using Go-diagrams and network models, they derive semialgebraic equations for each Deodhar component, enabling explicit computation of the bundle's boundary structure.
• They compute transition matrices between overlapping local trivializations of the fiber bundle and evaluate the sign of their determinants to test orientability.
• The analysis reveals that in approximately 3 out of 4 cases with $n \geq k+4$, the transition matrices have negative determinants, implying non-orientability.
• The authors conjecture that all such bundles are non-orientable, based on boundary structure analysis across positroid cells.

Experimental results

Research questions

• RQ1 Does the geometric space parametrized by Wilson loop diagrams in $\mathcal{N}=4$ SYM admit a global volume form?
• RQ2 To what extent is the fiber bundle over the positroid cell in the Grassmannian non-orientable?
• RQ3 Can the non-orientability be detected through the sign of transition function determinants in overlapping local trivializations?
• RQ4 How does the Deodhar decomposition refine the positroid stratification in capturing non-positive parts of the Grassmannian relevant to Wilson loops?
• RQ5 What is the role of characteristic classes in replacing volume forms when the underlying space is non-orientable?

Key findings

• The space parametrized by Wilson loop diagrams in $\mathcal{N}=4$ SYM is non-orientable in the majority of cases, particularly for $n \geq k+4$, due to fiber bundle transition functions with negative determinants.
• The non-orientability arises because the $4k$-dimensional fiber bundle over a positroid cell in $\mathrm{Gr}_{\mathbb{R}}(k,n+1)$ contains Deodhar components that lie outside the positive Grassmannian.
• A diagrammatic algorithm based on Go-diagrams is developed to read off the semialgebraic equations defining each Deodhar component explicitly.
• The top-dimensional Deodhar component in each fiber $\pi^{-1}(\Sigma(W))$ plays a role analogous to the positroid cell $\Sigma(W)$ in the positive Grassmannian.
• In approximately 3 out of 4 pairs $(k,n)$ with $n \geq k+4$, the bundle is non-orientable, and the authors conjecture this holds universally.
• Despite non-orientability, the authors argue that scattering amplitudes may still be captured by characteristic classes rather than volume forms, preserving the geometric program's viability.