Connections on non-abelian gerbes and their holonomy (Q2847370)

\textit{J.-L. Brylinski} [Loop spaces, characteristic classes and geometric quantization. Boston, MA: Birkhäuser (1993; Zbl 0823.55002)] introduced the notion of a connection on an abelian gerbe, showing that these represent classes in Deligne cohomology, which in degree two had been connected to two-dimensional conformal field theory in terms of the surface holonomy of a connection on an abelian gerbe by \textit{K. Gawȩdzki} in [``Topological actions in two-dimensional quantum field theories'', NATO ASI Ser., Ser. B, Phys. 185, 101--142 (1988; \url{doi:10.1007/978-1-4613-0729-7_5})]. For surface holonomy of connections on abelian gerbes one is referred to the second author's [IRMA Lect. Math. Theor. Phys. 16, 653--682 (2010; Zbl 1220.53034); Theory Appl. Categ. 18, 240--273 (2007; Zbl 1166.55005)]. NEWLINENEWLINENEWLINEWhile the notion of connections on non-abelian gerbes have already been discussed in [\textit{J. Fuchs} et al., in: European congress of mathematics. Proceedings of the 5th ECM congress, Amsterdam, Netherlands, July 14--18, 2008. Zürich: European Mathematical Society. 167--195 (2010; Zbl 1195.53043)] and [Zbl 1220.53034], its sincere developments have remained to be seen. The principal objective in the present paper is to present a general and systematic approach to connections on non-abelian gerbes together with notions of parallel transport and surface holonomy. The approach is general in the sense that it works for gerbes whose band is an arbitrary Lie \(2\)-groupoid and whose fibers are modelled by an arbitrary \(2\)-category. The approach is systematic in the sense that it is solely based on axioms for parallel transport along surfaces in terms of gluing laws and smoothness conditions.