Non-trivial bundles and defect operators in n-form gauge theories

In $(d+1)$-dimensional $1$-form non-Abelian gauge theories, we classify non-trivial $0$-form bundles in Rᵈ, which yield configurations of $D(d-2j)$-branes wrapping $(d-2j)$-cycles cd-2j in $Dd$-branes. We construct the related defect operators U⁽²ʲ⁻¹⁾ ( cd-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⁾ ( cd-2j ). When cd-2j is not exact, U⁽²ʲ⁻¹⁾ ( cd-2j ) and U⁽²ʲ⁻¹⁾ (- cd-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 non-trivial $1$-form bundles in $(d+1)$-dimensional $2$-form non-Abelian gauge theories, where the defect operators are U⁽²ʲ⁾ ( cd-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.