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.

We revisit constraints on decaying very heavy dark matter (VHDM) using the latest ultrahigh-energy cosmic-ray (UHECR; E >1e18 eV) data and ultrahigh-energy (UHE) gamma-ray flux upper limits, measured by the Pierre Auger Observatory. We present updated limits on the VHDM lifetime for masses up to ∼ 1e15 GeV, considering decay into quarks, leptons, and massive bosons. In particular, we consider not only the UHECR spectrum but their composition data that favors heavier nuclei. Such a combined analysis improves the limits at <1e12 GeV because VHDM decay does not produce UHECR nuclei. We also show that the constraints from the UHE gamma-ray upper limits are ∼ 10 times more stringent than that obtained from cosmic rays, for all of the Standard Model final states we consider. The latter improves our limits to VHDM lifetime by a factor of two for dark matter mass >1e12 GeV. We also provide constraints using neutrino flux from dark matter decay, including the neutrino-induced cascades. We consider the interaction of UHE neutrinos with the cosmic neutrino background, leading to the attenuation of the extragalactic flux reaching Earth, which improves our analysis to obtain tighter constraints.