Gauge transform

From DispersiveWiki

Many fields arising in PDE can be viewed as a section or connection on a gauge bundle, which is typically a principal G-bundle over a domain , where G is the gauge group. To (locally) coordinatize these sections and connections, one chooses a (local) trivialization of the gauge bundle, which identifies the bundle with the trivial bundle . This converts sections into G-valued fields , and connections D into -valued one-form , thus . Such a trivialization is known as a gauge.

Given any G-valued field U, one can transform the trivialization by applying the group element U(x) to the fiber of the trivial bundle at x. This is a gauge transform; it maps to and to .

One reason for applying a gauge transform is to convert a connection into a better form. However, there is an obstruction to flattening a connection entirely, namely the curvature of the connection. Nevertheless, there are a number of gauges which seek to make the connection as mild as possible given its curvature.

Examples of gauges

Equations in which gauge transforms arise