We show the explicit connection between two distinct and complementary approaches to the fractional quantum Hall system (FQHS): the quantum wires formalism and the topological low-energy effective description given in terms of an Abelian Chern-Simons theory. The quantum wires approach provides a description of the FQHS directly in terms of fermions arranged in an array of one-dimensional coupled wires. In this sense it is usually referred to as a microscopic description. On the other hand, the effective theory has no connection with the microscopic modes, involving only the emergent topological degrees of freedom embodied in an Abelian Chern-Simons gauge field, which somehow encodes the collective dynamics of the strongly correlated electrons. The basic strategy pursued in this work is to bosonize the quantum wires system and then consider the continuum limit. By examining the algebra of the bosonic operators in the Hamiltonian, we are able to identify the bosonized microscopic fields with the components of the field strength (electric and magnetic fields) of the emergent gauge field. Thus our study provides a bridge between the microscopic physical degrees of freedom and the emergent topological ones without relying on the bulk-edge correspondence. * Electronic address: weslei@uel.br † Electronic address: pedrogomes@uel.br ‡ Electronic address: hernaski@uel.br

We use Dirac matrix representations of the Clifford algebra to build fracton models on the lattice and their effective Chern-Simons-like theory. As an example, we build lattice fractons in odd D spatial dimensions and their (D+1) spacetime dimensional effective theory. The model possesses an anti-symmetric K matrix resembling that of hierarchical quantum Hall states. The gauge charges are conserved in sub-dimensional manifolds which ensures the fractonic behavior. The construction extends to any lattice fracton model built from commuting projectors and with tensor products of spin-1/2 degrees of freedom at the sites.

We derive an effective field theory for a type-II fracton starting from the Haah code on the lattice. The effective topological theory is not given exclusively in terms of an action; it must be supplemented with a condition that selects physical states. Without the constraint, the action only describes a type-I fracton. The constraint emerges from a condition that cube operators multiply to the identity, and it cannot be consistently implemented in the continuum theory at the operator level, but only in a weaker form, in terms of matrix elements of physical states. Informed by these studies and starting from the opposite end, i.e., the continuum, we discuss a Chern-Simons-like theory that does not need a constraint or projector, and yet has no mobile excitations. Whether this continuum theory admits a lattice counterpart remains unanswered.