On $3$-gauge transformations, $3$-curvature and $\mathbf{Gray}$-categories

Abstract

In the $3$-gauge theory, a $3$-connection is given by a $1$-form $A$ valued in the Lie algebra $\mathfrak{g}$, a $2$-form $B$ valued in the Lie algebra $\mathfrak{h}$ and a $3$-form $C$ valued in the Lie algebra $\mathfrak{l}$, where $(\mathfrak{g},\mathfrak{h},\mathfrak{l})$ 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 $2$-form, the $2$-curvature $3$-form and the $3$-curvature $4$-form. The gauge configurations can be interpreted as smooth $\mathbf{Gray}$-functors between two $\mathbf{Gray}$ $3$-groupoids: the path $3$-groupoid ${\mathcal{P}}_3(X)$ and the $3$-gauge group ${\mathcal{G}}^{\mathcal{L}}$ associated to the $2$-crossed module $\mathcal{L}$, whose differential is $(\mathfrak{g},\mathfrak{h},\mathfrak{l})$. The derivatives of $\mathbf{Gray}$-functors are $3$-connections, and the derivatives of lax-natural transformations between two such $\mathbf{Gray}$-functors are $3$-gauge transformations. We give the $3$-dimensional holonomy, the lattice version of the $3$-curvature, whose derivative gives the $3$-curvature $4$-form. The covariance of $3$-curvatures easily follows from this construction. This $\mathbf{Gray}$-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 $\{,\}$.

36 pages

References in corpus (2)

• Topological Higher Gauge Theory - from BF to BFCG theory
• M-brane Models from Non-Abelian Gerbes

Cited by in corpus (16)

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• Higher Gauge Theories Based on 3-groups
• Hamiltonian Analysis for the Scalar Electrodynamics as 3BF Theory
• Topological invariant of 4-manifolds based on a 3-group
• BCF anomaly and higher-group structure in the low energy effective theories of mesons
• 3-form Yang-Mills based on 2-crossed modules
• Gauge symmetry of the 3BF theory for a generic Lie 3-group
• On the global $2$-holonomy for a $2$-connection on a $2$-bundle
• Higher Chern-Simons based on (2-)crossed modules
• Henneaux-Teitelboim gauge symmetry and its applications to higher gauge theories
• Symmetry breaking mechanisms of the 3BF action for the Standard Model coupled to gravity
• The 3BF theory as a TQFT
• Correspondence between 3BF and Einstein-Cartan formulations of quantum gravity