The next example comes from gauge theory and to some extent is the non-abelian version of the previous example. The Yang-Mills equations for a connection on a principal bundle over a four-dimensional riemannian manifold say that the curvature 2-form

F_{A} = dA + ½[A,A]

has zero "divergence"

δ_{A} F_{A} ≔
⋆d_{A}⋆ F_{A} = 0,

where d_{A} is the exterior covariant derivative.
Notice that the Bianchi identity says that
d_{A} F_{A} = 0.

The Yang-Mills equation is second-order in the connection, which is the dynamical field in the theory, as the Aharanov-Bohm effect teaches us, instead of the curvature.

A special type of solutions to the Yang-Mills equation are the instantons. These are connections whose curvature is (anti)self-dual:

⋆ F_{A} = ±F_{A}.

For an instanton, the Yang-Mills equations automatically follow from the Bianchi identity. Notice that again the (anti)self-duality condition is first-order on the connection, whereas the Yang-Mills equation is second-order.

Furthermore instantons are very particular solutions, in that the minimise the Yang-Mills action:

∫ |F_{A}|^{2}.

In fact on a compact manifold this action is bounded below by a topological invariant of the bundle and the connections which satisfy the bound are precisely the instantons.

The underlying supersymmetric theory in this case is a euclidean version of four-dimensional N=2 supersymmetric Yang-Mills. The condition for a connection to preserve some supersymmetry is that its curvature should annihilate a nonzero spinor:

F_{A} ⋅ ε = 0,

where here the curvature two-form acts on spinors as an element of the Clifford algebra. It is a simple linear algebraic fact that a two-form annihilates a spinor if and only if it is (anti)self-dual.

This story has two natural extensions. First, some symmetric solutions of the four-dimensional Yang-Mills equation are actually solutions of the three-dimensional Yang-Mills-Higgs equation. The symmetric instantons give rise to the so-called BPS monopoles. This is the reason why states preserving some supersymmetry are called BPS-states. The underlying supersymmetric theory in this case is N=2 supersymmetric Yang-Mills.

The second natural extension is to higher dimensions, where it ties in with the ideas of generalised self-duality on manifolds possessing a co-closed 4-form ω. The "instantons" now obey

⋆ F_{A} = (⋆ω) ∧
F_{A},

where again the Bianchi identity implies the Yang-Mills equation.

For example, riemannian manifolds with special holonomy admit such co-closed four-forms. The four-dimensional case is the one where the four-form is the volume form, which requires no extra structure besides the orientation.

Supersymmetry also underlies this generalised self-duality. The relevant physical theory is now the maximal ten-dimensional supersymmetric Yang-Mills theory and its dimensional reductions.

© JosÃ© Figueroa-O'Farrill