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Super Wilson loops and holonomy on supermanifolds. (English) Zbl 1316.58004
In this interesting paper the author makes precise mathematical sense of super Wilson loops and the notion of holonomy for supermanifolds. The motivation is clearly rooted in super Yang-Mills theory. He does this by trying to follow more-or-less the classical constructions by employing the functor of points. In doing so the author presents the notion of a S-path (also see A. J. Bruce [Arch. Math., Brno 50, No. 2, 115–130 (2014; Zbl 1340.58002)]) using internal Homs objects in the category of supermanifolds. Loosely, S-paths are paths that are parametrised by external Grassmann odd parameters.
From there S-loops, connections, parallel transport and the holonomy of an S-point are developed in the ‘categorical language’. This leads to the notion of the holonomy group functor, which turns out not to be representable in general; we have a generalised supermanifold and not a genuine supermanifold. An Ambrose–Singer theorem is also established in this context.
Finally the relation with Galaev’s notion of holonomy for supermanifolds [A. S. Galaev, Abh. Math. Semin. Univ. Hamb. 79, No. 1, 47–78 (2009; Zbl 1182.58004)] is explored.

MSC:
58A50 Supermanifolds and graded manifolds
53C29 Issues of holonomy in differential geometry
18F05 Local categories and functors
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