# nLab D=4 Yang-Mills theory

Contents

### Context

#### Differential cohomology

differential cohomology

# Contents

## 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

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 $Sp(1)\rightarrow Sp(1),q\mapsto q^m$.

Last revised on June 28, 2024 at 14:20:59. See the history of this page for a list of all contributions to it.