On 3-gauge transformations, 3-curvatures, and Gray-categories

Abstract

In the 3-gauge theory, a 3-connection is given by a 1-form A valued in the Lie algebra g, a 2-form B valued in the Lie algebra h and a 3-form C valued in the Lie algebra l, where (g, h, l) constitutes a differential 2-crossed module. We give the 3-gauge transformations from a 3-connection to another, and show the transformation formulae of the 1-curvature 2form, the 2-curvature 3-form and the 3-curvature 4-form. The gauge configurations can be interpreted as smooth Gray-functors between two Gray 3-groupoids: the path 3-groupoid P3(X) and the 3-gauge group G L associated to the 2-crossed module L , whose differential is (g, h, l). The derivatives of Gray-functors are 3-connections, and the derivatives of lax-natural transformations between two such Gray-functors are 3-gauge transformations. We give the 3-dimensional holonomy, the lattice version of the 3-curvature, whose derivative gives the 3curvature 4-form. The covariance of 3-curvatures easily follows from this construction. This Gray-categorical construction explains why 3-gauge transformations and 3-curvatures have the given forms. The interchanging 3-arrows are responsible for the appearance of terms concerning the Peiffer commutator {, }.

Semistrict higher gauge theory

“…Deligne cohomology for categorified bundles was described previously in some special cases. In particular, the case of Abelian gerbes was discussed in [38], the case of principal 2-bundles with strict structure 2-group was given in [39], and the case of principal 3-bundles was presented in [10] (see also [40]). Here, we wish to describe the low-lying sets of the Deligne cohomology with values in a semistrict Lie 2-group.…”

Hamiltonian Analysis for the Scalar Electrodynamics as 3BF Theory

“…As was discussed in detail in [8], one formulates a topological 3BF action by specifying a particular gauge Lie 3-group. It has been proved that any strict 3-group is equivalent to a 2-crossed module [9,10].…”

“…see [9,10] for details. For this specific choice of a 3-group, where α = ω + A, given by the algebra-valued differential forms ω ∈ A…”

“…A testament to this is the level of complexity of the constrained 2BF formulation of general relativity [12], which is merely one sector in the action (12). Therefore, in this paper, we will limit ourselves to the topological sector of the theory, namely the unconstrained 3BF theory (9), which consists of the terms in the first row of Equation 12, and is written in full detail in Equation (10). One should be aware that this restriction changes various properties of the theory.…”

Division algebras and supersymmetry IV

“…Meanwhile, the mathematical theory of 3-connections is in its infancy. Their holonomy has been studied by Martins and Picken [31], and their gauge transformations by Saemann and Wolf [33] and in a different formulation by Wang [43]. Saemann and Wolf develop these ideas as an attack on six-dimensional superconformal field theories, which physicists will recognize as part of the program to develop M5-brane models in M-theory.…”