On higher holonomy invariants in higher gauge theory. I (Q2816550)

One of the basic problems in 3-dimensional topology is the classification of topologically distinct knots. Knots are closed embedded curves on a 3-fold \(M\). Analogously to common knots, surface knots are closed embedded surfaces in \(M\). Knot invariants take the same value for all topologically identical knots. Computing knot invariants, therefore, allows one to ascertain whether two given knots are topologically distinct or not. If they have different knot invariants, they are. Ordinary Chern-Simons theory allows in principle the computation of knot invariants using quantum field theory. The paper under review is the first 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.NEWLINENEWLINEThe goal of the paper is working out a rigorous definition of surface knot holonomy in strict higher gauge theory that is suitable for a gauge theoretic computation of surface knot invariants. Because of the nature of the approach in the paper under review, it is not required that knots or surface knots have codimension 2 in the ambient manifold. Therefore, the authors only assume that the dimension is sufficiently large to allow for embeddings of the relevant types. The interesting applications, of course, belong to the codimension 2 case. For a flat 2-connection, the definition of the 2-holonomy of surface knots of arbitrary genus is given and its covariance properties under 1-gauge transformation and change of base data is determined. A few simple sample computations to illustrate the abstract constructions is presented.NEWLINENEWLINEFor Part II see [the author, ibid. 13, No. 7, Article ID 1650091, 22 p. (2016; Zbl 1364.81197)].