Gauge theory studies principal bundle connections, called gauge fields, on a principal
bundle. These connections correspond to fields, in physics, such as an electromagnetic
field, and the Lie group of the principal
bundle corresponds to the symmetries of the physical system. The base
manifold to the principal bundle is usually
a four-dimensional manifold which corresponds to space-time.
In the case of an electromagnetic field, the symmetry group is the unitary
group .
The other two groups that arise in physical theories are the special
unitary groups
and . Also, a group
representation
of the symmetry group, called internal space, gives rise to an associated
vector bundle.

Actually,the principal bundle connections which minimize an energy functional are the only ones
of physical interest. For example, the Yang-Mills
connections minimize the Yang-Mills functional.
These connections are useful in low-dimensional
topology. In fact, in Donaldson theory, the
collection of Yang-Mills connections gives topological invariants of the base
manifold .

## See also

Donaldson Theory,

Group Representation,

Lie Group,

Manifold,

Principal Bundle,

Seiberg-Witten
Equations,

Vector Bundle
*This entry contributed by Todd
Rowland*

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## References

Friedman, R. and Morgan, J. W. (Eds.). *Gauge Theory and the Topology of Four-Manifolds.* Providence, RI: Amer. Math. Soc.,
1998.Naber, G. *Topology,
Geometry, and Gauge Fields.* New York: Springer-Verlag, 2000.## Referenced
on Wolfram|Alpha

Gauge Theory
## Cite this as:

Rowland, Todd. "Gauge Theory." From *MathWorld*--A Wolfram Web Resource, created by Eric
W. Weisstein. https://mathworld.wolfram.com/GaugeTheory.html

## Subject classifications