Foliated field theories and multipole symmetries

Due to the recent studies of the fracton topological phases, the host of which deconfined quasiparticle excitations with mobility restrictions, the concept of symmetries has been updated. Focusing on one of these new symmetries, multipole symmetries, including global, dipole, and quadruple symmetries, and gauge fields associated with them, we construct new sets of Z₍4pt{0ex} (2+1) -dimensional foliated background-field (BF) theories, where BF theories of conventional topological phases are stacked in layers with couplings between them. By investigating gauge-invariant nonlocal operators, we show that our foliated BF theories exhibit unusual ground-state degeneracy depending on the system size; it depends on the greatest common divisor between N and the system size. Our result provides an important connection between UV lattice models of the fracton topological phases and other unconventional ones in view of foliated field theories.