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A gauge theory may denote either a classical field theory or a quantum field theory whose field configurations are cocycles in differential cohomology (abelian or nonabelian).
An ordinary gauge theory is a quantum field theory whose field configurations are vector bundles with connection.
This includes notable the fields that carry the three fundamental forces of the standard model of particle physics:
Ordinary electromagnetism in the absence of magnetic charges is a gauge theory of $U(1)$-principal bundles with connection.
Fields in Yang-Mills theory (such as appearing in the standard model of particle physics and in GUTs) are vector bundles with connection.
Other examples include formal physical models.
The group $G$ in these examples is called the gauge group of the theory.
The above examples of gauge fields consisted of cocycles in degree-$1$ differential cohomology.
More generally, a higher gauge theory is a quantum field theory whose field configurations are cocycles in more general differential cohomology, for instance higher degree Deligne cocycles or more generally cocycles in other differential refinements, such as in differential K-theory.
This generalization does contain experimentally visible physics such as
But a whole tower of higher and generalized gauge theories became visible with the study of higher supergravity theories,
The Kalb-Ramond field is a bundle gerbe with connection, a Deligne cocycle with curvature 3-form.
The supergravity C-field is a Deligne cocycle with curvature 4-form.
The RR-field is a cocycle in differential K-theory.
There are various models that realize gravity also as a gauge theory.
In particular supergravity theories have interpretations as higher gauge theories as described at D'Auria-Fre formulation of supergravity.
Sometimes one see the view expressed that gauge symmetry is “just a redundancy” in the description of a theory of physics, for instance in that among observables it is only the gauge invariant ones which are physically meaningful.
This statement however
In the presence of magnetic charge (and then even in the absence of chiral fermion anomalies?) the standard would-be action functional for higher gauge theories may be ill-defined. The Green-Schwarz mechanism is a famous phenomenon in differential cohomology by which such a quantum anomaly cancels against that given by chiral fermions.
The following tries to give an overview of some collection of gauge fields in physics, their models by differential cohomology and further details.
cocycle in lowest degree nonabelian differential cohomology
originally realized in terms of differential Čech cocycles
with coefficients in the groupoid of Lie-algebra valued forms,
then traditionally in terms of vector bundles with connection
field strength depending on the group $G$ we have
$G = U(1)$ - electromagnetism (see below)
$G = SU(2)\times U(1)$ - electroweak force field strength
$G = SU(3)$ - strong nuclear force field
cocycle in degree-$2$ ordinary differential cohomology
field strength: the electric field $E$ and magnetic field $B$, locally at a point $x \in X$
on $X = \mathbb{R}^3\backslash \{0\}$: underlying class in integral cohomology $cl(\hat F) \in H(X,\mathbf{B} U(1)) \simeq H^2(X,\mathbb{Z})$ is the magnetic charge
parallel transport: gauge interaction piece of action functional of the electrically charged quantum 1-particle
cocycle in degree-$3$ ordinary differential cohomology
naturally/historically realized in terms of
a cocycle in Čech–Deligne cocycle
a bundle gerbe with connection
field strength: $H \in \Omega^3(X)$ the “$H$-field” – on a D-brane this is the magnetic current for the Yang-Mills field on the brane
parallel transport: gauge interaction piece of action functional of the electrically charged quantum 2-particle (the string).
cocycle in degree-$4$ ordinary differential cohomology
naturally/historically realized in terms of as a cocycle in Čech–Deligne cocycle
using the D'Auria-Fre formulation of supergravity it may also be thought of as a nonabelian differential cocycle given by a Cartan-Ehresmann ∞-connection
field strength: $H \in \Omega^4(X)$ the “$G$-field” – in heterotic supergravity this is the 5-brane magnetic current for the twisted Kalb-Ramond field
parallel transport: gauge interaction piece of action functional of the electrically charged quantum 3-particle (the membrane).
cocycle in differential K-theory
field strength: RR-forms
gauge field: models and components
An introduction to concepts in the quantization of gauge theories is in
A standard textbook on the BV-BRST formalism for the quantization of gauge systems is in
Discussion of abelian higher gauge theory in terms of differential cohomology is in
Dan Freed, Dirac charge quantization and generalized differential cohomology
Alessandro Valentino, Differential cohomology and quantum gauge fields (pdf)
José Figueroa-O’Farrill, Gauge theory (web page)
Tohru Eguchi, Peter Gilkey, Andrew Hanson, Gravitation, gauge theories and differential geometry, Physics Reports 66:6 (1980) 213—393 pdf
Discussion in the context of AQFT (or at least with aiming for such a formualtion) includes the following
and more specifically on the problem of locality.
An exposition of the relation to geometric Langlands duality is in
A discussion of “gauge” and gauge transformation in metaphysics is in
Hermann Weyl’s historical argument motivating gauge theory in physics from rescaling of units of length was given in 1918 in
Quick reviews include
More comprehensive historical accounts include
Lochlainn O'Raifeartaigh, The Dawning of Gauge Theory, Princeton University Press (1997)
Lochlainn O'Raifeartaigh, Norbert Straumann, Gauge Theory: Historical Origins and Some Modern Developments Rev. Mod. Phys. 72, 1-23 (2000).
Norbert Straumann, Gauge principle and QED, talk at PHOTON2005, Warsaw (2005) (arXiv:hep-ph/0509116)