Supersymmetry and gauge theory instantons

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

FA = dA + ½[A,A]

has zero "divergence"

δA FA ≔ ⋆dA⋆ FA = 0,

where dA is the exterior covariant derivative. Notice that the Bianchi identity says that dA FA = 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:

⋆ FA = ±FA.

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:

∫ |FA|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:

FA ⋅ 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

⋆ FA = (⋆ω) ∧ FA,

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.

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