nLab instanton sector

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.

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gauge field: models and components

physics	differential geometry	differential cohomology
gauge field	connection on a bundle	cocycle in differential cohomology
instanton/charge sector	principal bundle	cocycle in underlying cohomology
gauge potential	local connection differential form	local connection differential form
field strength	curvature	underlying cocycle in de Rham cohomology
gauge transformation	equivalence	coboundary
minimal coupling	covariant derivative	twisted cohomology
BRST complex	Lie algebroid of moduli stack	Lie algebroid of moduli stack
extended Lagrangian	universal Chern-Simons n-bundle	universal characteristic map