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 winning 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

Consider

• $G$ a Lie group

• with Lie algebra $\mathfrak{g}$,

• $\pi\colon P\twoheadrightarrow X$ a smooth principal $G$-bundle

• over a smooth manifold $X$.

Write

$$\mathcal{A} \;\coloneqq\; \Omega^1_{conn}(P;\mathfrak{g}) \;\subset\; \Omega^1(P;\mathfrak{g}) \,,$$

for the set of Lie algebra valued Ehresmann connection 1-forms on $P$.

This is an infinite-dimensional affine space, acted on by the group of gauge transformations

$$\mathcal{G} \;\coloneqq\; Aut(P) \simeq \Gamma\big(Ad(P)\big)$$

with quotient

$$\begin{array}{rcl} \mathcal{A} &\xrightarrow{\phantom{--}[-]\phantom{--}}& \mathcal{A}/\mathcal{G} \end{array}$$

The Yang-Mills equations $d_A \star F_A=0$, being gauge invariant, carve out a subspace of this quotient

$$\mathcal{M}_P \;\coloneqq\; \big\{ [A] \in \mathcal{A}/\mathcal{G} \,\big|\, d_A\star F_A=0 \big\}.$$

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:

$$\mathcal{M}_P^{\pm} \;=\; \;\coloneqq\; \big\{ [A] \in \mathcal{M}_P \;\big|\; \star F_A = \pm F_A \big\} \mathrlap{\,.}$$

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 $G$ being the second special unitary group SU(2) and $P$ a principal SU(2)-bundle: (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 $G$ being the third special orthogonal group SO(3) and $P$ a principal SO(3)-bundle: (Freed & Uhlenbeck 1991, Eq. (2.29) on p. 42 & eq. (10.3) on p. 155)

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

Examples

SU(2) Yang-Mills moduli spaces

For an oriented $4$-manifold $X$, principal SU(2)-bundles are fully classified by their second Chern class, meaning that $c_2\colon Prin_{SU(2)}(X)\rightarrow \mathbb{Z}$ is a bijection. For an integer $n\in\mathbb{Z}$, let $P_n\twoheadrightarrow X$ be the unique principal SU(2)-bundle with $c_2(P_c)=c$, then:

$$p_1 Ad(P_n)[X] =-4c_2(P_n)[X] =-4n.$$

(Donaldson & Kronheimer 97, Eq. (2.1.39)) (The adjoint representation $Ad\colon SU(2)\rightarrow SO(3)$ is the double cover.)

For $X=S^4$, one has signature $\sigma(S^4)=0$ and Euler characteristic $\chi(S^4)=2$. Hence the dimension formulas give:

$$dim\mathcal{M}_{S^4,P_n}^SDYM =-8n-3;$$

$$dim\mathcal{M}_{S^4,P_n}^ASDYM =-8n-3.$$

For $X=\mathbb{C}P^2$, one has intersection form $Q_{\mathbb{C}P^2}=(+1)$, signature $\sigma(Q_{\mathbb{C}P^2})=1$ and Euler characteristic $\chi(\mathbb{C}P^2)=3$. Hence the dimension formulas give:

$$dim\mathcal{M}_{\mathbb{C}P^2,P_n}^SDYM =-8n-3;$$

$$dim\mathcal{M}_{\mathbb{C}P^2,P_n}^ASDYM =-8n-6.$$

For $X=\overline{\mathbb{C}P^2}$, one has intersection form $Q_{\overline{\mathbb{C}P^2}}=(-1)$, signature $\sigma(Q_{\overline{\mathbb{C}P^2}})=-1$ and Euler characteristic $\chi(\overline{\mathbb{C}P^2})=3$. Hence the dimension formulas give:

$$dim\mathcal{M}_{\overline{\mathbb{C}P^2},P_n}^SDYM =8n-6;$$

$$dim\mathcal{M}_{\overline{\mathbb{C}P^2},P_n}^ASDYM =8n-3.$$

With $[\overline{\mathbb{C}P^2}]=-[\mathbb{C}P^2]$, there is an additional sign change compared to the previous example.

SO(3) Yang-Mills moduli spaces

For an oriented $4$-manifold $X$, principal SO(3)-bundles are fully classified by their second Stiefel-Whitney class and first Pontrjagin class, meaning that $(w_2,p_1[X])\colon Prin_{SO(3)}(X)\rightarrow H^2(X,\mathbb{Z}_2)\times\mathbb{Z}$ is a bijection. For a cohomology class $w\in H^2(X,\mathbb{Z}_2)$ and an integer $m\in\mathbb{Z}$, let $P_{c,m}\twoheadrightarrow X$ be the unique principal SO(3)-bundle with $w_2(P_{w,n})=w$ and $p_1(P_{w,n})[X]=n$, then:

$$p_1 Ad(P_{w,n})[X] =p_1(P_{w,n})[X] =n.$$

(The adjoint representation $Ad\colon SO(3)\rightarrow SO(3)$ is the identity.)

For $X=S^4$, one has signature $\sigma(S^4)=0$ and Euler characteristic $\chi(S^4)=2$. Hence the dimension formulas give:

$$dim\mathcal{M}_{S^4,P_{w,n}}^SDYM =2n-3;$$

$$dim\mathcal{M}_{S^4,P_{w,n}}^ASDYM =2n-3.$$

Related entries

On ordinary Yang-Mills theory (YM):

• D=2 YM, D=3 YM, D=4 YM, D=5 YM, D=6 YM, D=7 YM, D=8 YM

• Maxwell theory/electromagnetism (U(1) YM), Donaldson theory (SU(2) YM), quantum chromodynamics (SU(3) YM)

• Yang-Mills equation, linearized Yang-Mills equation, Yang-Mills instanton, Yang-Mills field, stable Yang-Mills connection, Yang-Mills moduli space, Yang-Mills flow, F-Yang-Mills equation, Bi-Yang-Mills equation

• Uhlenbeck's singularity theorem, Uhlenbeck's compactness theorem

On variants of Yang-Mills theory and on super Yang-Mills theory (SYM):

• Yang-Mills-Higgs equations, stable Yang-Mills-Higgs pair, Yang-Mills-Higgs flow

• Einstein-Yang-Mills theory, Einstein-Yang-Mills-Dirac theory, Einstein-Yang-Mills-Dirac-Higgs theory

• 3D Yang-Mills-Chern-Simons theory

• 3d superconformal gauge field theory: D=3 N=1 SYM, D=3 N=2 SYM, D=3 N=4 SYM

• 4d superconformal gauge field theory: D=4 N=1 SYM, D=4 N=2 SYM, D=4 N=4 SYM

• D=5 SYM, D=6 SYM, D=7 SYM, D=10 SYM

• topological Yang-Mills theory, topologically twisted D=4 super Yang-Mills theory

• 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)

• Simon Donaldson, Peter Kronheimer: The Geometry of Four-Manifolds (1990, revised 1997), Oxford University Press and Claredon Press, [oup:52942, doi:10.1093/oso/9780198535539.001.0001, ISBN:978-0198502692, ISSN:0964-9174]

See also:

• Wikipedia, Yang–Mills moduli space