Introduction to gauge theory

If we have a principle bundle whose base space is space or spacetime and structure group is a Lie group G, then, the space of smooth (although I do realize in physics, we often don't deal with smooth functions) sections of this bundle forms a group, called the group of gauge transformations. We can define a connection on this principle bundle, yielding a Lie algebra valued 1-form, A. From this 1-form, we can construct a Lie algebra valued 2-form, F by

${\mathbf {F}}=d{\mathbf {A}}+{\mathbf {A}}\wedge {\mathbf {A}}$

where d stands for the exterior derivative and $\wedge$ stands for the wedge product.

Infinitesimal gauge transformations form a Lie algebra, which is characterized by a smooth Lie algebra valued scalar, &epsilon. Under such an infinitesimal gauge transformation,

$\delta _{\epsilon }{\mathbf {A}}=[\epsilon ,{\mathbf {A}}]-d\epsilon$

where $[.,.]$ is the Lie product.

One thing to note is that not all gauge transformations can be generated by infinitesimal gauge transformations in general. For example, if the base manifold is a compact boundariless manifold such that the homotopy class of mappings from that manifold to G is nontrivial.

The Yang-Mills action is now given by

${\frac {1}{4g^{2}}}\int Tr[*F\wedge F]$

where * stands for the Hodge dual and the integral is defined as in differential geometry.

A quantity which is invariant under gauge transformations is the Wilson loop, which is defined over any closed path, &gamma, as follows:

$\chi ^{(\rho )}({\mathcal {P}}\{e^{\int _{\gamma }A}\})$

where &chi is the character of a complex representation &rho and ${\mathcal {P}}$ represents the path ordered operator.

The philosophy behind gauge theory

Basically, it states that symmetry transformations can only be performed locally. So, if you try to "rotate" something in a certain region, this does not determine how objects are rotated in another regions. So, the best way to summarize it is to say it is symmetry transformations are localized.

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