nLab Yang-Mills equation

Idea

The Yang-Mills equations are the equations of motion/Euler-Lagrange equations of Yang-Mills theory. They generalize Maxwell's equations.

Related concepts

• Yang-Mills instanton

• Yang-Mills monopole

References

(For full list of references see at Yang-Mills theory)

General

• Karen Uhlenbeck, notes by Laura Fredrickson, Equations of Gauge Theory, lecture at Temple University, 2012 (pdf, pdf)

• DispersiveWiki, Yang-Mills equations

• TP.SE, Which exact solutions of the classical Yang-Mills equations are known?

Solutions

General

Wu and Yang (1968) found a static solution to the sourceless $SU(2)$ Yang-Mills equations. Recent references include

• J. A. O. Marinho, O. Oliveira, B. V. Carlson, T. Frederico, Revisiting the Wu-Yang Monopole: classical solutions and conformal invariance

There is an old review,

• Alfred Actor, Classical solutions of $SU(2)$ Yang—Mills theories, Rev. Mod. Phys. 51, 461–525 (1979),

that provides some of the known solutions of $SU(2)$ gauge theory in Minkowski (monopoles, plane waves, etc) and Euclidean space (instantons and their cousins). For general gauge groups one can get solutions by embedding $SU(2)$‘s.

Instantons and monopoles

For Yang-Mills instantons the most general solution is known, first worked out by

• Michael Atiyah, Nigel Hitchin, Vladimir Drinfeld, Yuri Manin, Construction of instantons, Physics Letters 65 A, 3, 185–187 (1978) pdf

for the classical groups SU, SO , Sp, and then by

• C. Bernard, N. Christ, A. Guth, E. Weinberg, Pseudoparticle Parameters for Arbitrary Gauge Groups, Phys. Rev. D16, 2977 (1977)

for exceptional Lie groups. The latest twist on the Yang-Mills instanton story is the construction of solutions with non-trivial holonomy:

• Thomas C. Kraan, Pierre van Baal, Periodic instantons with nontrivial holonomy, Nucl.Phys. B533 (1998) 627-659, hep-th/9805168

There is a nice set of lecture notes

• David Tong, TASI Lectures on Solitons (hep-th/0509216),

on topological solutions with different co-dimension (Yang-Mills instantons, Yang-Mills monopoles, vortices, domain walls). Note, however, that except for instantons these solutions typically require extra scalars and broken U(1)‘s, as one may find in super Yang-Mills theories.

Some of the material used here has been taken from

• TP.SE, Which exact solutions of the classical Yang-Mills equations are known?

Another model featuring Yang-Mills fields has been proposed by Curci and Ferrari, see Curci-Ferrari model.