The Twofold Way of Super Holonomy

There are two different notions of holonomy in supergeometry, the supergroup introduced by Galaev and our functorial approach motivated by super Wilson loops. Either theory comes with its own version of invariance of vectors and subspaces under holonomy. By our first main result, the Twofold Theorem, these definitions are equivalent. Our proof is based on the Comparison Theorem, our second main result, which characterises Galaev's holonomy algebra as an algebra of coefficients, building on previous results. As an application, we generalise some of Galaev's results to S-points, utilising the holonomy functor. We obtain, in particular, a de Rham-Wu decomposition theorem for semi-Riemannian S-supermanifolds.Comment: 25 pages, in this second version many improvements mainly thanks to suggestions by Alexander Alldridge and Dominik Ostermay

Vertex Operators of Super Wilson Loops

We study the supersymmetric Wilson loop as introduced by Caron-Huot, which attaches to lightlike polygons certain edge and vertex operators, whose shape is determined by supersymmetry constraints. We state explicit formulas for the vertex operators to all orders in the Gra{\ss}mann expansion, thus filling a gap in the literature. This is achieved by deriving a recursion formula out of the supersymmetry constraints.Comment: 11 page

Holomorphic Supercurves and Supersymmetric Sigma Models

We introduce a natural generalisation of holomorphic curves to morphisms of supermanifolds, referred to as holomorphic supercurves. More precisely, supercurves are morphisms from a Riemann surface, endowed with the structure of a supermanifold which is induced by a holomorphic line bundle, to an ordinary almost complex manifold. They are called holomorphic if a generalised Cauchy-Riemann condition is satisfied. We show, by means of an action identity, that holomorphic supercurves are special extrema of a supersymmetric action functional.Comment: 30 page

Compactness for Holomorphic Supercurves

We study the compactness problem for moduli spaces of holomorphic supercurves which, being motivated by supergeometry, are perturbed such as to allow for transversality. We give an explicit construction of limiting objects for sequences of holomorphic supercurves and prove that, in important cases, every such sequence has a convergent subsequence provided that a suitable extension of the classical energy is uniformly bounded. This is a version of Gromov compactness. Finally, we introduce a topology on the moduli spaces enlarged by the limiting objects which makes these spaces compact and metrisable.Comment: 38 page