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p-form symmetry
p-form symmetry in physics is a generalization of the concept of symmetry. An ordinary, continuous symmetry is associated with conserved charge, which is typically an integral over whole of space. In D spatial dimension, the conserved charge of continuous p-form symmetries is an integral over D-p subspace of the D-dimensional space, i.e. a flux conservation. Such a charge is both conserved under the deformation of the D-p dimensional subspace as well as in time.

A prototypical example of a 1-form symmetry in three spatial dimensions is the magnetic flux conservation and the Gauss' law in electromagnetism (although in electromagnetism the Gauss' law is an emergent 1-form symmetry, and is not exact because of the presence of dynamical electrically charged matter). The conserved magnetif flux through a surface $$\Sigma$$ is defined by

$$\Phi_\Sigma=\int_{\Sigma} d\vec S\cdot \vec B$$

The above flux is conserved in the sense that it is the same even if we deform the surface $$\Sigma$$. This follows from the fact that any change of the above integral from such a deformation is an integral over a contractible surface, and since a contractible surface can be written as a boundary of some 3-dimensional volume. By Gauss' theorem, this change is then written as a 3-dimensional integral over $$\nabla\cdot\vec B$$ which vanishes by Maxwell's equations. Further one can also establish that the time derivative of the flux is zero by using $$\partial_t\vec B=-\nabla\times \vec E$$ and the Stokes' theorem.

A similar analysis holds for the electric field, however the Maxwell's equations now indicate that electric charges and currents can spoil the invariance of the flux. However since all particles carrying electric charge are massive, such currents are absent in the vacuum of the theory, and the Gauss's law holds.