nLab Yang-Mills moduli space

Idea

The Yang-Mills moduli space (short YM moduli space, also instanton moduli space) is the moduli space of the Yang-Mills equations, hence the space of its solutions up to gauge. It is used in the proof of Donaldson's theorem, which was listed as Simon Donaldson‘s contributions for einning the Fields medal, and to defined the Donaldson invariants used to study 4-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, important subspaces of the Yang-Mills moduli space are the self-dual Yang-Mills moduli space (short SDYM moduli space, also self-dual instanton moduli space) of solutions of the self-dual Yang-Mills equations up to gauge and the anti self-dual Yang-Mills moduli space (short ASDYM moduli space, also anti self-dual instanton moduli space) of solutions of the anti self-dual Yang-Mills equations up to gauge. See also D=4 Yang-Mills theory and self-dual Yang-Mills theory.

Definition

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and $\pi\colon P\twoheadrightarrow X$ be a principal $G$-bundle over a smooth manifold $X$, which automatically makes $P$ a smooth manifold as well. Let $Ad(P)\coloneqq P\times_G\mathfrak{g}$ be the adjoint bundle, then the Yang-Mills equations as well as the (anti) self-dual Yang-Mills equations are formulated on the configuration space:

$$\mathcal{A} =\Omega^1(X,Ad(P)) \cong\Omega_{Ad}^1(P,\mathfrak{g})$$

where the isomorphism requires a choice of local sections $s_i\colon U_i\hookrightarrow P$ for an open cover $(U_i)_{i\in I}\subset X$ (or alternatively a connection since the latter space is an affine vector space, which makes the isomorphism non-canonical) and is then given by:

$$\Omega_{Ad}^1(P,\mathfrak{g})\xrightarrow\cong\Omega^1(X,Ad(P)), A\mapsto(s_i^*A)_{i\in I}.$$

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:

$$\mathcal{G} \coloneqq C^\infty(P,G)^G \cong C^\infty(X,G) \cong Aut(P)$$

where the isomorphisms are given using the free and transitive action of $G$ on the fibers of $P$ (with $G$ as a superscript meaning the $G$-equivariant maps and which are canonical):

$$C^\infty(X,G)\xrightarrow\cong Aut(P), f\mapsto(p\mapsto p\cdot f(\pi(p)));$$

$$C^\infty(P,G)^G\xrightarrow\cong Aut(P), f\mapsto(p\mapsto p\cdot f(p)).$$

A principal bundle automorphism $P\rightarrow P$ induces a vector bundle automorphism $Ad(P)\rightarrow Ad(P)$, causing the gauge group $\mathcal{G}$ to act free on the configuration space $\mathcal{A}$ and resulting in the orbit space:

$$\mathcal{B} \coloneqq\mathcal{A}/\mathcal{G}.$$

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:

$$\mathcal{M}_P\colon \coloneqq\{[A]\in\mathcal{A}|d_A\star F_A=0\}.$$

If $X$ is a 4-manifold, then D=4 Yang-Mills theory furthermore allows the definition of the (anti) self-dual Yang-Mills moduli space:

$$\mathcal{M}_P^- =\mathcal{M}_P^\mathrm{ASD}\colon \coloneqq\{[A]\in\mathcal{A}|d_AF_A=-F_A\};$$

$$\mathcal{M}_P^+ =\mathcal{M}_P^\mathrm{SD}\colon \coloneqq\{[A]\in\mathcal{A}|\star F_A=+F_A\}.$$

There are canonical inclusions $\mathcal{M}_P^-,\mathcal{M}_P^+\hookrightarrow\mathcal{M}_P^+$. The intersection $\mathcal{M}_P^-\cap\mathcal{M}_P^+$ includes exactly the flat connections, the critical points of the Chern-Simons action functional, and could therefore be refered to as Chern-Simons moduli space.

Properties

If $X$ is a 4-manifold, then: (Donaldson 1983, p. 290, Donaldson 1987, Eq. (2.1), Freed & Uhlenbeck 1991, Eq. (2.28))

$$dim\mathcal{M}_P^+ =2p_1(Ad(P))[X] -\dim G(1-b_1+b_2^-).$$

In particular for $G=\operatorname{SU}(2)$ the second special unitary group: (Freed & Uhlenbeck 1991, between eq. (2.28) and (2.29) on p. 42)

$$dim\mathcal{M}_P^+ =-8c_2(P)[X] -3(1-b_1+b_2^-).$$

In particular for $G=\operatorname{SO}(3)$ the third special orthogonal group: (Freed & Uhlenbeck 1991, Eq. (2.29) on p. 42 & eq. (10.3) on p. 155)

$$dim\mathcal{M}_P^+ =2p_1(P)[X] -3(1-b_1+b_2^-).$$

Related entries

• Seiberg-Witten moduli space (monopole moduli space)

References

• Simon Donaldson. An application of gauge theory to four-dimensional topology. In: Journal of Differential Geometry. 18. Jahrgang, Nr. 2, 1. Januar 1983, doi:10.4310/jdg/1214437665
• Simon Donaldson. The orientation of Yang-Mills moduli spaces and 4-manifold topology. In: Journal of Differential Geometry. 26. Jahrgang, Nr. 3, 1. Januar 1987, doi:10.4310/jdg/1214441485
• Daniel Freed, Karen Uhlenbeck, Instantons and Four-Manifolds, Mathematical Sciences Research Institute Publications, Springer 1991 (doi:10.1007/978-1-4613-9703-8)

See also:

• Wikipedia, Yang–Mills moduli space