Strings as Hyper-Fractons

We systematically examine all possible Gauss laws obeying spatial rotation symmetry, characterising the corresponding conserved charges. In the case of conserved higher moments, this gives rise to fractonic behaviour. We show that many Gauss laws, including those arising from $p$-form electrodynamics, in fact, produce an infinite tower of conserved moments, which we dub hyper-fractonic. In hyper-fractonic systems, a finite number of charged particles cannot be mobile due to an inability of fulfilling the infinite number of conservation laws with a finite number of degrees of freedom. Instead, mobile charged objects must have an infinite number of degrees of freedom. In particular, the strings and branes naturally coupling to $p$-form potentials provide an example of hyper-fractonic matter.