The Twofold Way of Super Holonomy

TL;DR

This paper establishes the equivalence of two different notions of super holonomy in supergeometry, introduces a comparison theorem, and applies these results to generalize existing theorems and decompose supermanifolds.

Contribution

It proves the equivalence of Galaev's supergroup holonomy and a functorial approach, and develops a comparison theorem linking their holonomy algebras.

Findings

Proves the equivalence of two super holonomy notions.

Develops a comparison theorem for holonomy algebras.

Generalizes Galaev's results to S-points and establishes a de Rham-Wu decomposition.

Abstract

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.