# Gauge transform

(Redirected from Gauge transformation)

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 ${\displaystyle \Omega }$, 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 ${\displaystyle \Omega \times G}$. This converts sections into G-valued fields ${\displaystyle \sigma }$, and connections D into ${\displaystyle {\mathfrak {g}}}$-valued one-form ${\displaystyle A}$, thus ${\displaystyle D_{\alpha }=\partial _{\alpha }+A_{\alpha }}$. 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 ${\displaystyle \sigma }$ to ${\displaystyle U\sigma }$ and ${\displaystyle A_{\alpha }}$ to ${\displaystyle UA_{\alpha }U^{-1}-\partial _{\alpha }UU^{-1}}$.

One reason for applying a gauge transform is to convert a connection ${\displaystyle A_{\alpha }}$ 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.