We extend recent work on hydrodynamics with global multipolar symmetries-known as "fracton hydrodynamics"-to systems in which the multipolar symmetries are gauged. We refer to the latter as "fracton magnetohydrodynamics", in analogy to magnetohydrodynamics (MHD), which governs systems with gauged charge conservation. Fracton MHD arises naturally from higher-rank Maxwell's equations and in systems with one-form symmetries obeying certain constraints; while we focus on "minimal" higher-rank generalizations of MHD that realize diffusion, our methods may also be used to identify other, more exotic hydrodynamic theories (e.g., with magnetic subdiffusion). In contrast to phenomenological MHD, our approach-based on higher-form symmetries-provides a systematic treatment of hydrodynamics beyond "weak coupling", and provides an effective hydrodynamic description, e.g., of quantum spin liquids, whose emergent gauge fields are strongly coupled.
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