Yang-Mills theory



physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics

Quantum field theory

Differential cohomology



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

1g 2 Xtr(F F )+iθ Xtr(F F ) \nabla \mapsto \frac{1}{g^2 }\int_X tr(F_\nabla \wedge \star F_\nabla) \;+\; i \theta \int_X tr(F_\nabla \wedge F_\nabla)



Classification of solutions


Despite its fundamental role in the standard model of particle physics, various details of the quantization of Yang-Mills theory are still open. See at quantization of Yang-Mills theory.


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



Yang-Mills theory is named after the article

which was the first to generalize the principle of electromagnetism to a non-abalian gauge group. This became accepted as formulation of QCD and weak interactions (only) after spontaneous symmetry breaking (the Higgs mechanism) was understood in the 1960s.

For modern reviews of the basics see

Lecture notes include

See also the references at QCD, gauge theory, and super Yang-Mills theory.

Classical discussion of YM-theory over Riemann surfaces (which is closely related to Chern-Simons theory, see also at moduli space of flat connections) is in

  • Michael Atiyah, Raoul Bott, The Yang-Mills equations over Riemann surfaces, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 308, No. 1505 (Mar. 17, 1983), pp. 523-615 (jstor, lighning summary)

which is reviewed in the lecture notes

For the relation to instanton Floer homology see also

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

For the relation to Tamagawa numbers see

  • Aravind Asok, Brent Doran, Frances Kirwan, Yang-Mills theory and Tamagawa numbers (arXiv:0801.4733)

Classical solutions

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

that provides some of the known solutions of SU(2)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)SU(2)‘s. For instantons the most general solution is known, first worked out by

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 groups. The latest twist on the 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

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

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

See also

Revised on January 16, 2017 14:25:54 by Urs Schreiber (