nLab D=4 Yang-Mills theory

Idea

D=4 Yang-Mills theory (D=4 YM theory) studies the Yang-Mills equations over a base manifold of dimension $D=4$. This special case allows to reduce the Yang-Mills equations of second order to the (anti) self-dual Yang-Mills equations ((A)SDYM equations) of first order.

Basics

Consider

• $G$ a Lie group,

• $B$ an orientable Riemannian 4-manifold,

• $E\twoheadrightarrow B$ a principal $G$-bundle,

• $A\in\Omega_{\operatorname{Ad}}^1(E,\mathfrak{g})\cong\Omega^1(B,\operatorname{Ad}(E))$ a principal connection,

• $F_A \coloneqq \mathrm{d}_A A=\mathrm{d}A+[A\wedge A]\in\Omega_{\operatorname{Ad}}^2(E,\mathfrak{g})\cong\Omega^2(B,\operatorname{Ad}(E))$ its curvature.

Chern-Weil theory implies that the second Chern class of the gauge bundle is:

(1)$$\big\langle c_2(E),[B] \big\rangle = \Big\langle c_2\big(\operatorname{Ad}(E)\big), [B] \Big\rangle = -\frac{1}{8\pi^2} \int_B \operatorname{tr}(F_A \wedge F_A) \;\in\; \mathbb{Z} \,.$$

Application on the 4-sphere

The quaternionic Hopf fibration is a principal $SU(2)$-bundle over $S^4$, which encodes the charge quantization of the magnetic charge of a magnetic monopole in five dimensions (Wu-Yang monopole) using:

$$\operatorname{Prin}_{\operatorname{Sp}(1)}(S^4) \;\cong\; \big[S^4,\operatorname{BSp}(1)\big] \;=\; \pi_4\operatorname{BSp}(1) \;\cong\; \pi_3\operatorname{Sp}(1) \;\cong\; \pi_3S^3 \;\cong\; \mathbb{Z} \,.$$

Given an $m\in\mathbb{Z}$, the corresponding principal bundle is given by pullback of the universal principal bundle $ESp(1)\twoheadrightarrow BSp(1)$ along the composition of the canonical inclusion $S^4\cong\mathbb{H}P^1\hookrightarrow\mathbb{H}P^\infty\cong BSp(1)$ and the map $BSp(1)\rightarrow BSp(1)$ induced by $Sp(1)\rightarrow Sp(1),q\mapsto q^m$.

Related concepts

• D=2 Yang-Mills theory
• D=5 Yang-Mills theory
• D=4 N=1 super Yang-Mills theory
• D=4 N=2 super Yang-Mills theory
• D=4 N=4 super Yang-Mills theory
• topologically twisted D=4 super Yang-Mills theory