Types of quantum field thories
FQFT and cohomology
Yang–Mills theory is a gauge theory on a given 4-dimensional (pseudo-)Riemannian manifold whose field is the Yang–Mills field – a cocycle in differential nonabelian cohomology represented by a vector bundle with connection – and whose action functional is
the Hodge star operator of the metric ;
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
which is reviewed in the lecture notes
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 Yang-Mills equations. Recent references include
There is an old review,
that provides some of the known solutions of 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 ‘s. For instantons the most general solution is known, first worked out 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.