Yang–Mills moduli space

In gauge theory, the Yang–Mills moduli space is the moduli space of the Yang–Mills equations, hence the space of its solutions up to gauge. It is used in Donaldson's theorem, proven in and improved in, which was listed as a contribution for Simon Donaldson winning the Fields Medal in 1986, and to defined the Donaldson invariants used to study four-dimensional smooth manifolds. A difficulity is, that the Yang–Mills moduli space is usually not compact and has to be compactified around singularities through laborious techniques. An improvement later appeared with the always compact Seiberg–Witten moduli space. The Yang–Mills moduli space is named after Chen-Ning Yang and Robert Mills, who introduced the underlying Yang–Mills equations in 1954.
In four dimensions, see also four-dimensional Yang–Mills theory, important subspaces of the Yang-Mills moduli space are the self-dual Yang-Mills moduli space of solutions of the self-dual Yang-Mills equations up to gauge and the anti self-dual Yang-Mills moduli space of solutions of the anti self-dual Yang-Mills equations up to gauge.

Definition

Let be a Lie group with Lie algebra and be a principal -bundle over a smooth manifold, which automatically makes a smooth manifold as well. Let be the adjoint bundle, then the Yang–Mills equations as well as the self-dual Yang–Mills equations are formulated on the configuration space:
where the isomorphism requires a choice of local sections for an open cover and is then given by:
Since the configuration space is an infinite-dimensional vector space, it is more difficult to handle. But also due to the group action on the principal bundle, it is plausible to consider a group action on the configuration space with the following gauge group:
where the isomorphisms are given using the free and transitive action of on the fibers of :
A principal bundle automorphism induces a vector bundle automorphism, causing the gauge group to act free on the configuration space and resulting in the orbit space:
It can be shown that the Yang–Mills equations are gauge invariant and hence are formulated over just this orbit space. Its solution form the Yang–Mills moduli space:
If is a 4-manifold, then four-dimensional Yang–Mills theory furthermore allows the definition of the self-dual Yang–Mills moduli space:
There are canonical inclusions. The intersection includes exactly the flat connections, the critical points of the Chern–Simons action functional, and could therefore be referred to as Chern–Simons moduli space.

Properties

The anti self-dual Yang–Mills moduli space of a principal SU(2)-bundle over a Riemannian 4-manifold is orientable. This more generally holds for principal -bundles.
The self-dual Yang–Mills moduli space of a principal -bundle with over a compact orientable Riemannian 4-manifold is compact.
If is a 4-manifold, then:

It is important to consider the different sign conventions regarding the characteristic classes in the first term: Instantons and Four-Manifolds uses the convention here, while the later The Geometry of Four-Manifolds uses the reverse convention.

Application

Self-dual SU(2) moduli space

For the proof of Donaldson's theorem, Simon Donaldson considered the self-dual Yang–Mills moduli space of the unique principal SU(2)-bundle with over a simply connected Riemannian 4-manifold with negative definite intersection form. After first assuming simple connectedness for in, he expanded the proof to also work without it in. If the intersection form is definite, then furthermore. According to the above formula, the self-dual Yang–Mills moduli space is five-dimensional. Simon Donaldson then gave the following description of its singularities and its boundary, resulting in a bordism essential for his proof:

If there are pairs to the equation with the Kronecker pairing and fundamental class, then there are singularities, so that is a 5-manifold. Every such singularity has a neighborhood diffeomorphic to a cone over the second complex projective space .
There exists a neighborhood of the boundary which is diffeomorphic to a half-open cylinder over, meaning that yields a compactification with. If additionally the cones around the singularities are removed, then describes a bordism between and. It is important, that all second complex projective spaces have the same orientation, since is nullbordant and hence could be canceled out. Since the signature is invariant under bordisms and transfers disjoint unions into sums, one has.

Self-dual SO(3) moduli space

Ron Fintushel and Ronald J. Stern considered the self-dual Yang–Mills moduli space of a principal -bundle with over a simply connected Riemannian 4-manifold with negative definite intersection form. As before, and. According to the above formula, the self-dual Yang–Mills moduli space is one-dimensional.