Gauge Theory

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 U(1)={e^(itheta)}. The other two groups that arise in physical theories are the special unitary groups SU(2) and SU(3). Also, a group representation V 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 M.

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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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.

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Gauge Theory

Cite this as:

Rowland, Todd. "Gauge Theory." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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