In quantum field theory, conserved particle numbers are associated with ordinary global symmetries. However, many theories can also possess a conserved density of extended objects such as strings, branes, etc. The generalized symmetry principle associated with such conservation laws is just as powerful as that for ordinary symmetries but has only recently been systematically explored. I will explain some of the resulting insights, discussing the classification of low-energy phases and discussing the emergence of gapless Goldstone modes when they are spontaneously broken. Such a symmetry also plays an important role in characterizing the long-distance physics of familiar Maxwell electrodynamics in four dimensions; as an application, I will discuss the realization of this symmetry at finite temperature and provide a reformulation of magnetohydrodynamics from the point of view of symmetry and effective field theory.
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