On higher holonomy invariants in higher gauge theory. II. (Q2816551)

This is the second of a series of two technical papers devoted to the analysis of holonomy invariants in strict higher gauge theory with end applications in higher Chern-Simons theory. In the first paper of a series of two, the authors pointed out that the success of 3-dimensional Chern-Simons theory as a quantum field theoretic framework for the computation of ordinary knot invariants suggests that a 4-dimensional version of Chern-Simons theory may do the same with regard to surface knot invariants.NEWLINENEWLINEThe higher dimensional analogs of plane Chern-Simons theory exist only in odd dimensional spaces, the realization of Chern-Simons theory appropriate for surface knots is likely to belong to the domain of higher gauge theory. Characterization of the topology of knotted surfaces in 4-dimensions by means of suitable higher invariants, is the higher dimensional analogue to the topological classification of ordinary knots in 3-dimensions through their invariants.NEWLINENEWLINEIn ordinary Chern-Simons theory, the computation of knot invariants involves the evaluation of traces of Wilson loops of the gauge field, mathematically holonomies of the gauge connection, along knots in representations of the gauge group. In a strict higher gauge theory, the corresponding issue for surface knots has two parts: (a) the definition of surface holonomies and the analysis of their dependence on the choice of gauge and base data; (b) the definition of the appropriate notion of trace for the gauge crossed module yielding surface knot invariants upon application to surface holonomies. The first part has been treated in the first paper of the series and the second one is the topic of the present second paper under review.NEWLINENEWLINEFor Part I see [the author, ibid. 13, No. 7, Article ID 1650090, 59 p. (2016; Zbl 1364.81196)].