# nLab instanton sector

cohomology

### Theorems

#### $\infty$-Chern-Weil theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

# Contents

## Definition

### In general

Given a gauge field configuration modeled by a $G$-principal connection, its instanton sector or charge sector is the equivalence class of the underlying principal bundle.

### In $SU(n)$-Yang-Mills theory

Notably for Yang-Mills theory on a 4-dimensional spacetime and with a gauge group the special unitary group $G = SU(n)$, $G$-principal bundles $P$ are entirely classified by their second Chern class $c_2(P)$ and hence the value $c_2(P) \in H^4(X, \mathbb{Z})$ is the instanton sector. Given the $G$-principal connection of the gauge field the image in de Rham cohomology of this class may be expressed by the integration of differential forms $[\int_{X} \langle F_\nabla , \F_\nabla \rangle] \in H_{dR}^4(X)$, where $F_{\nabla}$ is the curvature and $\langle -,-\rangle$ the invariant polynomial which corresponds to $c_2$ under the Chern-Weil homomorphism.

gauge field: models and components

Revised on June 10, 2013 15:20:36 by Urs Schreiber (89.204.137.122)