Nontrivial bundles and defect operators in $n$-form gauge theories

In $(d+1)$-dimensional $1$-form nonabelian gauge theories, we classify nontrivial $0$-form bundles in $\mathbb{R}^{d}$, which yield configurations of $D(d-2j)$-branes wrapping $(d-2j)$-cycles $c_{d-2j}$ in $Dd$-branes. We construct the related defect operators $U^{(2j-1)} ( c_{d-2j} )$, which are disorder operators carrying the $D(d-2j)$ charge. We compute the commutation relations between the defect operators and Chern-Simons operators on odd-dimensional closed manifolds, and derive the generalized Witten effect for $U^{(2j-1)} ( c_{d-2j} )$. When $c_{d-2j}$ is not exact, $U^{(2j-1)} ( c_{d-2j} )$ and $U^{(2j-1)} (- c_{d-2j} )$ can also combine into an electric $(2j-1)$-form global symmetry operator, where the $(2j-1)$-form is the Chern-Simons form. The dual magnetic $(d-2j)$-form global symmetry is generated by the $D(d-2j)$ charge. We also study nontrivial $1$-form bundles in $(d+1)$-dimensional $2$-form nonabelian gauge theories, where the defect operators are $\mathcal{U}^{(2j)} ( c_{d-2j-1} )$. With the field strength of the $1$-form taken as the flat connection of the $2$-form, we classify the topological sectors in $2$-form theories.