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An ordinary gauge field (such as the electromagnetic field or the fields that induce the nuclear force) is a field (in the sense of physics) which is locally represented by a differential 1-forms (the “gauge potential”) and whose field strength is, locally a differential 2-form. For instance for the electromagnetic field this differential 2-form is the Faraday tensor.
Roughly speaking, a higher gauge field is similarly a field which is locally represented by differential forms of higher degree.
One “reason” why an ordinary gauge field has a gauge potential given, lcoally, by a differential 1-forms $A$ is that the trajectory of a charged particle is a 1-dimensional curve in spacetime $X$, its worldline, hence a smooth function $\gamma \colon \Sigma_1 \to X$, and that the canonical way to produce an action functional on the mapping space of such curves is the integration of 1-forms over curves:
This is the parallel transport map or the holonomy map, if $\Sigma_1$ is a closed manifold. The contribution to the Euler-Lagrange equation of the particle obtained from the variation of this action functional is the Lorentz force which is exerrted by the background gauge field on the particle.
When one generalizes in this picture from 0-dimesional particles with 1-dimensional worldlines to $p$-dimensional particles (often called “p-branes”) with $(p+1)$-dimensional worldvolumes $\gamma_{p+1} \colon \Sigma_{p+1} \to X$, then one needs, locally, a differential (p+1)-form $A_{p+1}$ on spacetime $X$
The field strength or flux of such a higher gauge field is, accordingly, locally the $(p+2)$-form $F_{p+2}$.
The archetypical example of such a higher gauge field is the (hypothetical) Kalb-Ramond field or B-field (a precursor of the axion field under KK-compactification) to which the charged 1-brane, the “string”, couples. This is locally a differential 2-form $B_2$, and the gauge-coupling term in the action functional for the string is accordinly, locally, of the form
It keeps going this way: next one may consider “2-branes”, i.e. membranes and these will couple to a 3-form gauge field. For instance the membrane which gives the name to M-theory (the M2-brane) couples to a 3-form field called the supergravity C-field.
But there is an important further aspect to higher gauge fields, which makes this simple picture of higher degree differential forms drastically more rich:
Where an ordinary gauge field has gauge transformations $A_1 \mapsto A'_1$ given, locally, by smooth functions (0-forms) $\lambda_0$ via the de Rham differential $d_{dR}$
so a higher gauge field has, locally, higher gauge transformations given by $p$-forms $\lambda_p$:
But for $p \geq 0$ then a crucial new effect appears: these gauge transfomrations, being higher differential forms themselves have “gauge-of-gauge transformations” between them, given by lower degree forms.
This phenomenon implies that higher gauge fields have a rich global (“topological”) structure, witnessed by the higher analog of their instanton sectors. Namely while a higher gauge field to which a p-brane may couple is locally given by a $(p+1)$-form $A_{p+1}$, as one moves across coordinate charts this form gauge transforms by a $p$-form, which then itself, as one passes along two charts, transforms by a $(p-1)$-form, and so ever on.
The global structure for higher gauge fields obtained by carrying out this globalization via higher gauge transformations is the higher analog of that of a fiber bundle with connection on a bundle in higher differential geometry. This is sometimes known as a gerbe or, more generally, a principal infinity-bundle.
In fact the situation that there is just one gauge potential of degree $(p+1)$ with field strength of degree $(p+2)$ is just the simplest case, the “ordinary” case. More abstractly one says that such higher gauge fields are cocycles in ordinary differential cohomology.
More generally it may happen in higher gauge theory that the gauge potential is a formal linear combination of differential forms in various degrees.
The canonical example of this phenomenon is the RR-field in string theory. This has, locally, a gauge potential being a differential form in every even degree, or every odd degree. If one is careful about the higher gauge transformations in this situation to find the correct global structre (the “instanton sector”) structure of the higher gauge field, then one finds that this now is a cocycle in a differential generalized cohomology, namley in what is called differential topological K-theory. This may be understood as a higher and generalized form of the famous Dirac charge quantization condition for the electromagnetic field, see Freed 00. A lot of the fine detail of the anomaly cancellation in type II string theory depends on being careful about the global nature of this K-theoretic higher gauge RR-field (Distler-Freed-Moore 09)
Introduction and exposition includes
John Baez, John Huerta, An invitation to higher gauge theory, General Relativity and Gravitation 43 (2011), 2335-2392 (arXiv:1003.4485)
PhysicsForums-Insights Why higher category theory in physics?
Examples of prequantum field theories II: Higher gauge fields
For technical introduction to the RR-field as a higher gauge field see
Daniel Freed, Dirac charge quantization and generalized differential cohomology, Surveys in Differential Geometry, Int. Press, Somerville, MA, 2000, pp. 129–194 (arxiv:hep-th/0011220)
Jacques Distler, Dan Freed, Greg Moore, Orientifold Précis in: Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory Proceedings of Symposia in Pure Mathematics, AMS (2011) (arXiv:0906.0795, slides)
For foundations of higher prequantum field theory see
For foundations of higher gauge theory formalized in homotopy type theory see