Nontrivial bundles and defect operators in n-form gauge theories

A bstract In ( d + 1)-dimensional 1-form nonabelian gauge theories, we classify nontrivial 0-form bundles in ℝ d , which yield configurations of D ( d − 2 j )-branes wrapping ( d − 2 j )-cycles c d −2 j in Dd -branes. We construct the related defect operators U (2 j −1) ( c d −2 j ), which are disorder operators carrying the D ( d – 2 j ) 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 (2 j −1) ( c d −2 j ). When c d −2 j is not exact, U (2 j −1) ( c d −2 j ) and U (2 j −1) (– c d −2 j ) can also combine into an electric (2 j – 1)-form global symmetry operator, where the (2 j – 1)-form is the Chern-Simons form. The dual magnetic ( d – 2 j )-form global symmetry is generated by the D ( d – 2 j ) charge. We also study nontrivial 1-form bundles in ( d + 1)-dimensional 2-form nonabelian gauge theories, where the defect operators are $$ {\mathcal{U}}^{(2j)}\left({c}_{d-2j-1}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>U</mml:mi> <mml:mfenced> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>j</mml:mi> </mml:mrow> </mml:mfenced> </mml:msup> <mml:mfenced> <mml:msub> <mml:mi>c</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> <mml:mi>j</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mfenced> </mml:math> . 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.