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Fractons emerge as charges with reduced mobility in a new class of gauge theories. Here, we generalise fractonic theories of $U(1)$ type to what we call $(k,n)$-fractonic Maxwell theory, which employs symmetric order-$n$ tensors of $k$-forms (rank-$k$ antisymmetric tensors) as vector potentials. The generalisation has two key manifestations. First, the objects with mobility restrictions extend beyond simple charges to higher order multipoles (dipoles, quadrupoles, $ldots$) all the way to $n^mathrm{th}$-order multipoles. Second, these fractonic charges themselves are characterized by tensorial densities of $(k-1)$-dimensional extended objects. The source-free sector exhibits `photonic excitations with dispersion $omegasim q^n$.
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