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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 $\nabla \in \mathbf{H}(X,\bar \mathbf{B}U(n))$ in differential nonabelian cohomology represented by a vector bundle with connection – and whose action functional is
for
$F_\nabla$ the field strength, locally the curvature $\mathfrak{u}(n)$-Lie algebra valued differential form on $X$ ( with $\mathfrak{u}(n)$ the Lie algebra of the unitary group $U(n)$);
$\star$ the Hodge star operator of the metric $g$;
$\frac{1}{g^2}$ the Yang-Mills coupling constant and $\theta$ the theta angle, some real numbers (see at S-duality).
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
Arthur Jaffe, Edward Witten, Quantum Yang-Mills theory (pdf)
Simon Donaldson, Yang-Mills theory and geometry (2005) pdf
See also the references at QCD, gauge theory, and super Yang-Mills theory.
For the relation to instanton Floer homology see also
For the relation to Tamagawa numbers see
Wu and Yang (1968) found a static solution to the sourceless $SU(2)$ Yang-Mills equations. Recent references include
There is an old review,
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. For instantons the most general solution is known, first worked out by
for the classical groups SU, SO , Sp, and then by
for exceptional groups. The latest twist on the instanton story is the construction of solutions with non-trivial holonomy:
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.