  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.
