The Twofold Way of Super Holonomy
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.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2015
- DOI:
- 10.48550/arXiv.1505.04056
- arXiv:
- arXiv:1505.04056
- Bibcode:
- 2015arXiv150504056G
- Keywords:
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- Mathematical Physics;
- High Energy Physics - Theory;
- Mathematics - Algebraic Geometry;
- Mathematics - Differential Geometry;
- 58A50;
- 53C29;
- 18F05
- E-Print:
- 25 pages, in this second version many improvements mainly thanks to suggestions by Alexander Alldridge and Dominik Ostermayr