Field Theory for Superconducting Branes and Generalized Particle-Vortex Duality

We propose a field theory of closed $p$-brane $C_p^{}$ interacting with a $(p+1)$-form gauge field $A_{p+1}^{}$. This is a generalization of the Ginzburg-Landau theory (Abelian-Higgs model) for superconducting particles to higher-dimensional superconducting branes. A higher-form gauge invariant action is constructed by utilizing the Area derivative, which is a higher-dimensional generalization of the ordinary derivative. We find that the fundamental phenomena of superconductivity, such as the Meisser effect, topological defects, topological order, are naturally extended in the brane-field theory. We explicitly construct a topologically non-trivial static configuration that is characterized by the first homotopy group. Then, we calculate the low-energy effective theory in the presence of the topological defect and find that it is described by a BF-type topological field theory coupled with the world-volume of the topological defect. We also conjecture an infra-red duality between the superconducting brane-field model and dual brane-field model with a global $\mathrm{U}(1)$ higher-form symmetry as a generalization of the Particle-Vortex duality.