[Paper Review] Super Wilson Loops and Holonomy on Supermanifolds

Josua Groeger|arXiv (Cornell University)|Dec 17, 2013

Black Holes and Theoretical Physics18 references3 citations

TL;DR

This paper develops a supergeometric model of super Wilson loops on supermanifolds using S-points and Graßmann generators to encode supersymmetry, introducing a novel supergeometric parallel transport that defines holonomy as a Lie group-valued functor. The key contribution is a new Ambrose-Singer theorem and holonomy principle for supermanifolds, showing that Galaev's holonomy superalgebra can be extracted from this functorial holonomy via geometric coefficient extraction, though the two theories are not equivalent.

ABSTRACT

The classical Wilson loop is the gauge-invariant trace of the parallel transport around a closed path with respect to a connection on a vector bundle over a smooth manifold. We build a precise mathematical model of the super Wilson loop, an extension introduced by Mason-Skinner and Caron-Huot, by endowing the objects occurring with auxiliary Graßmann generators coming from S-points. A key feature of our model is a supergeometric parallel transport, which allows for a natural notion of holonomy on a supermanifold as a Lie group valued functor. Our main results for that theory comprise an Ambrose-Singer theorem as well as a natural analogon of the holonomy principle. Finally, we compare our holonomy functor with the holonomy supergroup introduced by Galaev in the common situation of a topological point. It turns out that both theories are different, yet related in a sense made precise.

Motivation & Objective

• To construct a mathematically rigorous model of super Wilson loops using S-points and Graßmann generators to capture supersymmetric structure.
• To define a supergeometric parallel transport on supermanifolds, enabling a natural notion of holonomy as a Lie group-valued functor.
• To establish a supermanifold analog of the Ambrose-Singer theorem and the holonomy principle.
• To compare the proposed functorial holonomy with Galaev's holonomy supergroup in the case of topological points.
• To determine whether Galaev's holonomy algebra can be recovered from the new holonomy functor in a purely algebraic way.

Proposed method

• Model super Wilson loops using S-points with auxiliary Graßmann generators to encode supersymmetry, inspired by 'maps with flesh' in superfield theory.
• Define supergeometric parallel transport via morphisms from S×[0,1] to a supermanifold M, generalizing classical path-ordered exponentials.
• Construct holonomy as a functor from the category of S-points to the category of Lie groups, assigning to each S-point the group of parallel transport automorphisms.
• Derive a super Ambrose-Singer theorem by expressing the holonomy Lie algebra in terms of curvature components evaluated at S-points.
• Use higher-order derivatives of holonomy elements at special S-points (e.g., T-points) to extract generators of Galaev's holonomy superalgebra.
• Apply differential calculus on supermanifolds to compute covariant derivatives of curvature and relate them to holonomy via path-ordered exponentials and pullbacks.

Experimental results

Research questions

• RQ1 How can super Wilson loops be rigorously modeled using supergeometry and S-points with Graßmann generators?
• RQ2 What is the correct generalization of parallel transport and holonomy to supermanifolds, and how can it be formulated as a functor?
• RQ3 Can a super Ambrose-Singer theorem be established, relating holonomy Lie algebra to curvature at S-points?
• RQ4 How does the proposed holonomy functor relate to Galaev’s holonomy supergroup in the case of topological points?
• RQ5 Can the generators of Galaev’s holonomy superalgebra be extracted from the new holonomy functor in a purely algebraic way?

Key findings

• A new super Ambrose-Singer theorem is established, expressing the holonomy Lie algebra as the span of curvature components evaluated at S-points.
• A natural analogon of the classical holonomy principle is proven, linking parallel sections to holonomy-invariant vectors in the super setting.
• The proposed holonomy functor is not representable, so it differs fundamentally from Galaev’s holonomy supergroup in the common case of topological points.
• Generators of Galaev’s holonomy superalgebra can be extracted from the new holonomy functor by taking coefficients of specific elements in the T-point construction, based on geometric significance.
• The extraction process is geometric rather than purely algebraic, leaving open the question of whether a purely algebraic reconstruction is possible.
• The model successfully reproduces the characteristic formulas of super Wilson loops from [7], validating its physical relevance through mathematical consistency.