In physics and mathematics, and especially differential geometry and gauge theory, the Yang–Mills equations are a system of partial differential equations for a connection on a vector bundle or principal bundle.
In their foundational paper on the topic of gauge theories, Robert Mills and Chen-Ning Yang developed (essentially independent of the mathematical literature) the theory of principal bundles and connections in order to explain the concept of gauge symmetry and gauge invariance as it applies to physical theories.
[2] The novelty of the work of Yang and Mills was to define gauge theories for an arbitrary choice of Lie group
corresponding to electromagnetism, and the right framework to discuss such objects is the theory of principal bundles.
One assumes that the fundamental description of a physical model is through the use of fields, and derives that under a local gauge transformation (change of local trivialisation of principal bundle), these physical fields must transform in precisely the way that a connection
The first attempt at choosing a canonical connection might be to demand that these forms vanish.
On a principal bundle the correct way to phrase this condition is that the curvature
In this sense they are the natural choice of connection on a principal or vector bundle over a manifold from a mathematical point of view.
, so Yang–Mills connections can be seen as a non-linear analogue of harmonic differential forms, which satisfy In this sense the search for Yang–Mills connections can be compared to Hodge theory, which seeks a harmonic representative in the de Rham cohomology class of a differential form.
in this affine space, the curvatures are related by To determine the critical points of (3), compute The connection
classifies all connections modulo gauge transformations, and the moduli space
Moduli spaces of Yang–Mills connections have been intensively studied in specific circumstances.
Michael Atiyah and Raoul Bott studied the Yang–Mills equations for bundles over compact Riemann surfaces.
This is the Narasimhan–Seshadri theorem, which was proved in this form relating Yang–Mills connections to holomorphic vector bundles by Donaldson.
[5] In this setting the moduli space has the structure of a compact Kähler manifold.
The moduli space of ASD connections, or instantons, was most intensively studied by Donaldson in the case where
[Note 1] For various choices of principal bundle, one obtains moduli spaces with interesting properties.
These spaces are Hausdorff, even when allowing reducible connections, and are generically smooth.
may be extended across the point at infinity using Uhlenbeck's removable singularity theorem.
Using analytical results of Clifford Taubes and Karen Uhlenbeck, Donaldson was able to show that in specific circumstances (when the intersection form is definite) the moduli space of ASD instantons on a smooth, compact, oriented, simply-connected four-manifold
in two ways: once using that signature is a cobordism invariant, and another using a Hodge-theoretic interpretation of reducible connections.
Interpreting these counts carefully, one can conclude that such a smooth manifold has diagonalisable intersection form.
The moduli space of ASD instantons may be used to define further invariants of four-manifolds.
Dimensional reduction is the process of taking the Yang–Mills equations over a four-manifold, typically
[13] Hitchin showed the converse, and Donaldson proved that solutions to the Nahm equations could further be linked to moduli spaces of rational maps from the complex projective line to itself.
[14][15] The duality observed for these solutions is theorized to hold for arbitrary dual groups of symmetries of a four-manifold.
Indeed there is a similar duality between instantons invariant under dual lattices inside
[3] Symmetry reductions of the ASD equations also lead to a number of integrable systems, and Ward's conjecture is that in fact all known integrable ODEs and PDEs come from symmetry reduction of ASDYM.
For example reductions of SU(2) ASDYM give the sine-Gordon and Korteweg–de Vries equation, of
In this case the moduli space admits a geometric quantization, discovered independently by Nigel Hitchin and Axelrod–Della Pietra–Witten.