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 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 ${\mathcal{U}}^{\left(2j\right)}\left(c_{d-2j-1}\right)$ \( {\mathcal{U}}^{(2j)}\left({c}_{d-2j-1}\right) \) . 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.