Effective Brane Field Theory with Higher-form Symmetry

We propose an effective field theory for branes with higher-form symmetry as a generalization of ordinary Landau theory, which is an extension of the previous work by Iqbal and McGreevy for one-dimensional objects to an effective theory for $p$-dimensional objects. In the case of a $p$-form symmetry, the fundamental field $\psi [C_p^{}]$ is a functional of $p$-dimensional closed brane $C_p^{}$ embedded in a spacetime. As a natural generalization of ordinary field theory, we call this theory the brane field theory. In order to construct an action that is invariant under higher-form transformation, we generalize the idea of area derivative for one-dimensional objects to higher-dimensional ones. Following this, we discuss various fundamental properties of the brane field based on the higher-form invariant action. It is shown that the classical solution exhibits the area law in the unbroken phase of $\mathrm{U}(1)$ $p$-form symmetry, while it indicates a constant behavior in the broken phase for the large volume limit of $C_p^{}$. In the latter case, the low-energy effective theory is described by the $p$-form Maxwell theory. We also discuss brane-field theories with a discrete higher-form symmetry and show that the low-energy effective theory becomes a BF-type topological field theory, resulting in topological order. Finally, we present a concrete brane-field model that describes a superconductor from the point of view of higher-form symmetry.