group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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.
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.
non-perturbative effect, non-perturbative quantum field theory, non-perturbative string theory
string theory FAQ – Isn’t it fatal that the string perturbation series does not converge?
gauge field: models and components