A topological viewpoint on consistency aspects of electromagnetic theory on a lattice

In many applications, the continuum electromagnetic (EM) theory needs to be replaced by a discrete theory on a spatial lattice. A lattice EM theory has well-known phenomena not present on the continuum counterpart, such as finite frequency cut-off and rotational symmetry breaking. These may be viewed as approximations of to the continuum theory, but not as inconsistencies. The latter usually leads to pervasive and more deleterious phenomena such as a spurious model and unconditional instabilities. The authors explore the natural path to find a lattice approximation to a field theory formulated in terms of differential forms. This will employ the known mapping of forms onto linear functions (cochains) on the space of some lattice elements (chains). Then, we use the natural factorization of Maxwell's equations (MEs) into a topological part and a metric part to describe the exact (combinatorial) lattice counterparts to the topological equations and the nature of the lattice approximation to the metric equations. The objective is to discuss basic requirements for an ab initio consistent lattice EM theory, clarifying their relationship with topological and metric aspects of the theory.