The authors are interested in studying some particular geometric aspects in the context of so-called 2-bundles, i.e., bundles of categories. In particular, they address the differential geometry of categorical group \(2\)-bundles over a smooth manifold \(M\) and their two-dimensional parallel transport, in the context of cubical sets and to define so-called Wilson surface observables. Following the approach of N. Hitchin (2001) and M. Mackaay and R. F. Picken (2002) they consider a coordinate neighborhood description of \(2\)-bundles with connection. (They refer to previous works on \(2\)-bundles by D. M. Roberts and U. Schreiber (2008), T. Bartles (2004–2006), C. Wockel (2008), R. Brown and P. J. Higgins (1978, 1981) and R. Sivera (2010).)
One of the main results is that the \(2\)-dimensional holonomy of a \(2\)-bundle connection does not depend on the chosen coordinate neighborhoods (up to very simple transformations). Let \({\mathcal G}\equiv(\partial:E\to G,\triangleleft)\) be a Lie crossed module, where \(\triangleleft\) is a left action of \(G\) on \(E\) by automorphism. They show that the cubical \({\mathcal G}-2\) bundle holonomy, associated to oriented embedded \(2\)-spheres \(\Sigma\subset M\), yield an element \({\mathcal W}({\mathcal B},\Sigma)\in\ker\partial\subset E\) (the Wilson sphere observable) independent of the parametrization of the spheres and the chosen coordinate neighborhoods, but depend only on the equation class of the cubical \({\mathcal G}\)-\(2\) bundle with connection \({\mathcal B}\). They show also that a holonomy can still be defined for non-sphere surfaces embedded in \(M\) even if it depends on the isotopy type of the parametrization. (An example with Wilson tori is considered.)
The authors claim that the main motivation for this study is related to the problem of obtaining invariants of knotted surfaces in \(S^4\) via categorified gauge actions. They emphasize the analogy with Witten’s three-dimensional approach which used Wilson loop observables and the Chern-Simons action (1989) and led to a physical definition of the Jones polynomial. (They also recall Vassiliev invariants as the M. Kontsevich integral (1993), the configuration space integral of Bott and Taubes, and invariants of knotted surfaces by A. S. Cattaneo and C. A. Rossi (2005).)
The paper after a careful introduction, splits into four more sections with a detailed index reported just after the abstract.
Reviewer’s mark. This paper fits in the so-called “higher gauge theory”, that aims to translate the standard formalism of Gauge Theory, where point particles move by parallel transport, under connections on bundles, to extended objects (as considered in string theory). This suggested some authors to categorify concepts from topology and geometry, just replacing smooth manifolds by smooth categories, Lie groups by Lie \(2\)-groups, etc. In this respect, let us emphasize that even if this categorical translation appears completely inadequate to encode very complex physical phenomena like quantum field theory and quantum (super)gravity, genuine motivations for such generalizations may be found in themselves, as natural efforts to recognize more general mathematical structures. Really, in order to codify dynamics of quantum field theory the right category is the one of quantum (super)manifolds, as introduced by the reviewer of this paper. There one can recognize that also “string theory” finds its natural dynamical description in the framework of the geometric theory of quantum (super) PDEs. (See the following paper: arXiv:0906.1363, and works by the same author on quantum PDEs quoted there.)