Gauge transform

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

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

Examples of gauges

• Coulomb gauge
• Temporal gauge
• Lorenz gauge
• Cronstrom gauge
• Radial gauge
• Caloric gauge

Equations in which gauge transforms arise

• Benjamin-Ono equation
• cubic DNLS on R
• Einstein equations
• Maxwell-Klein-Gordon equations
• Schrodinger maps
• Yang-Mills equations
• Wave maps