( k , n ) -fractonic Maxwell theory

Fractons emerge as charges with reduced mobility in a class of gauge theories. Here, we generalize fractonic theories of U(1) type to what we call (k,n)-fractonic Maxwell theory, which employs symmetric rank-n tensors of k forms (rank-k antisymmetric tensors) as "vector potentials." The generalization, valid in any spatial dimension d, has two key manifestations. First, the objects with mobility restrictions extend beyond simple charges to higher-order multipoles (dipoles, quadrupoles, etc.) all the way to (n−1)th-order multipoles, which we call the order-n fracton condition. Second, these fractonic charges themselves are characterized by tensorial densities of (k−1)-dimensional extended objects. For any (k,n), the theory can be constructed to have a gapless "photon modes" with dispersion ω∼|q|z, where the integer z can range from 1 to n.