nLab
Yang-Mills theory

Idea

Yang–Mills theory is a gauge theory on a given 4-dimensional (pseudo-)Riemannian manifold $X$ whose field is the Yang–Mills field – a cocycle $\nabla \in H(X,\overline{B}U(n))$ in differential nonabelian cohomology represented by a vector bundle with connection – and whose action functional is

$$\nabla \mapsto \frac{1}{e^2}{\int }_X{F}_{\nabla }\wedge \star F_{\nabla }+i\theta {\int }_X{F}_{\nabla }\wedge F_{\nabla }$$\nabla \mapsto \frac{1}{e^2 }\int_X F_\nabla \wedge \star F_\nabla + i \theta \int_X F_\nabla \wedge F_\nabla

for

• $F_{\nabla }$ the field strength, locally the curvature $𝔲(n)$-Lie algebra valued differential form on $X$ ( with $𝔲(n)$ the Lie algebra of the unitary group $U(n)$);

• $\star$ the Hodge star operator of the metric $g$;

• $\frac{1}{e^2}$ and $\theta$ some real numbers (see S-duality)

Applications

All gauge fields in the standard model of particle physics as well as in GUT models are Yang–Mills fields.

The matter fields in the standard model are spinors charged under the Yang-Mills field. See

• spinors in Yang-Mills theory

Related concepts

• standard model of particle physics

  • electromagnetism

  • spinors in Yang-Mills theory

  • QED, QCD

• super Yang-Mills theory

• S-duality, Montonen-Olive duality

  • electric-magnetic duality

  • geometric Langlands duality

• Chern-Simons theory

• Yang-Mills instanton

References

General

• Arthur Jaffe?, Edward Wittem?, Quantum Yang-Mills theory (pdf)

• Simon Donaldson, Yang-Mills theory and geometry (2005) (pdf)

For the relation to instanton Floer homology see also

• Simon Donaldson, Floer homology groups in Yang-Mills theory Cambridge University Press (2002) (pdf)

Classical solutions

Wu and Yang (1968) found a static solution to the sourceless $\mathrm{SU}(2)$ 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 $\mathrm{SU}(2)$ Yang—Mills theories , Rev. Mod. Phys. 51, 461–525 (1979),

that provides some of the known solutions of $\mathrm{SU}(2)$ 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 $\mathrm{SU}(2)$’s. For instantons the most general solution is known, first worked out by

• Michael Atiyah, Nigel Hitchin, Vladimir Drinfeld, Yuri Manin

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

• Bernard, Christ, Guth, Weinberg

for exceptional groups. The latest twist on the instanton story is the construction of solutions with non-trivial holomony?:

& Kraan, van Baal, Periodic instantons with nontrivial holonomy. , Nucl.Phys. B533 (1998) 627-659.

There is a nice set of lecture notes

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

on topological solutions with different co-dimension (instantons, 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?