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There are several distinct frameworks within which higher gauge theories have been developed. Alvarez et al. [1] extend the notion of integrability to higher dimensions in the context of geometric field theories. Several works [2] of John Baez, Urs Schreiber and coauthors have developed higher gauge theories heavily based on category theory.
Here the mathematical study of gauge theory overlaps significantly with its physical origins, as the standard model of particle physics can be thought of as a quantum field theory on a four-dimensional spacetime. The study of gauge theory problems in four dimensions naturally leads to the study of topological quantum field theory. Such theories ...
A particular choice of the scalar and vector potentials is a gauge (more precisely, gauge potential) and a scalar function ψ used to change the gauge is called a gauge function. [ citation needed ] The existence of arbitrary numbers of gauge functions ψ ( r , t ) corresponds to the U(1) gauge freedom of this theory.
In this case, the quiver gauge theory is a four-dimensional = supersymmetric gauge theory. The quiver gauge theory in higher dimensions can be defined similarly. The quiver is particularly convenient for representing conformal gauge theory.
In mathematical physics, two-dimensional Yang–Mills theory is the special case of Yang–Mills theory in which the dimension of spacetime is taken to be two. This special case allows for a rigorously defined Yang–Mills measure, meaning that the (Euclidean) path integral can be interpreted as a measure on the set of connections modulo gauge transformations.
Gauge symmetry is an example of a local symmetry, with the symmetry described by a Lie group (which mathematically describe continuous symmetries), which in the context of gauge theory is called the gauge group of the theory. Quantum chromodynamics and quantum electrodynamics are famous examples of gauge theories.
Electromagnetic theory possess the simplest kind of local gauge symmetry called () (see unitary group). A theory that displays local gauge invariance is called a gauge theory. In order to formulate other gauge theories we turn the above reasoning inside out. This is the subject of the next section.
The concept and the name of gauge theory derives from the work of Hermann Weyl in 1918. [1] Weyl, in an attempt to generalize the geometrical ideas of general relativity to include electromagnetism, conjectured that Eichinvarianz or invariance under the change of scale (or "gauge") might also be a local symmetry of general relativity.