Brane field theory with higher-form symmetry

We propose field theory for branes with higher-form symmetry as a generalization of ordinary Landau theory. The field \psi[C_p^{}] becomes a functional of p-dimensional closed brane Cp embedded in a spacetime. As a natural generalization of ordinary field theory, we call this theory brane field theory. In order to construct an action that is invariant under higher-form transformation, we first generalize the concept of “derivative” for higher-dimensional objects. Then, 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 U(1) p-form symmetry, while it indicates a constant behavior in the broken phase for the large volume limit of Cp. In the latter case, the low-energy effective theory is described by the p-form Maxwell theory. If time permits, 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.