Yang–Mills–Higgs equations

In mathematics, the Yang–Mills–Higgs equations are a set of non-linear partial differential equations for a Yang–Mills field, given by a connection, and a Higgs field, given by a section of a vector bundle. These equations are

${\displaystyle {\begin{aligned}D_{A}*F_{A}+[\Phi ,D_{A}\Phi ]&=0,\\D_{A}*D_{A}\Phi &=0\end{aligned}}}$

with a boundary condition

${\displaystyle \lim _{|x|\rightarrow \infty }|\Phi |(x)=1}$

where

A is a connection on a vector bundle,

D_A is the exterior covariant derivative,

F_A is the curvature of that connection,

Φ is a section of that vector bundle,

∗ is the Hodge star, and

[·,·] is the natural, graded bracket.

These equations are named after Chen Ning Yang, Robert Mills, and Peter Higgs. They are very closely related to the Ginzburg–Landau equations, when these are expressed in a general geometric setting.

M.V. Goganov and L.V. Kapitanskii have shown that the Cauchy problem for hyperbolic Yang–Mills–Higgs equations in Hamiltonian gauge on 4-dimensional Minkowski space have a unique global solution with no restrictions at the spatial infinity. Furthermore, the solution has the finite propagation speed property.