  We propose nonabelian higher-rank gauge theories in 2+1D and 3+1D. The gauge
group is constructed from the volume-preserving diffeomorphisms of space. We
show that the intriguing physics of the lowest Landau level (LLL) limit can be
interpreted as the consequences of the symmetry. We derive the renowned
Girvin-MacDonald-Platzman (GMP) algebra as well as the topological Wen-Zee term
within our formalism. Using the gauge symmetry in 2+1D, we derive the LLL
effective action of vortex crystal in rotating Bose gas as well as Wigner
crystal of electron in an applied magnetic field. We show that the non-linear
sigma models of ferromagnet in 2+1D and 3+1D exhibit the higher-rank gauge
symmetries that we introduce in this paper. We interpret the fractonic behavior
of the excitations on the lowest Landau level and of skyrmions in ferromagnets
as the consequence of the higher-rank gauge symmetry.
