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.

Details

General form

(…)

Abelian case

For an abelian Lie group as structure group, its Lie algebra is also abelian and hence all Lie brackets vanish and makes the Yang-Mills equation reduce to the Maxwell equation:

$$\mathrm{d}\star\mathrm{d}A \;=\; 0 \,.$$

Properties

Relation to generalized Laplace equation

Let:

$$\Delta_A \;\coloneqq\; \delta_A\mathrm{d}_A \,+\, \mathrm{d}_A \delta_A \;\colon\; \Omega^k\big( B,\, Ad(E) \big) \longrightarrow \Omega^k\big( B,\, Ad(E) \big)$$

be a generalized Laplace operator.

The Bianchi identity $\mathrm{d}_A F_A=0$ and the Yang-Mills equation $\delta_A F_A=0$ combine to:

$$\Delta_A F_A \;=\; 0 \,.$$

Related concepts

• Yang-Mills instanton

• Yang-Mills monopole

• F-Yang-Mills equation

• Bi-Yang-Mills equation

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.