Continuum limit for a discrete Hodge–Dirac operator on square lattices

We study the continuum limit for Dirac-Hodge operators defined on the
n dimensional square lattice hZ^n as h goes to 0. This result extends to a first order
discrete differential operator the known convergence of discrete Schrödinger operators
to their continuous counterpart. To be able to define such a discrete analog, we start
by defining an alternative framework for a higher–dimensional discrete differential
calculus. We believe that this framework, that generalize the standard one defined on
simplicial complexes, could be of independent interest. We then express our operator
as a differential operator acting on discrete forms to finally be able to show the limit to
the continuous Dirac-Hodge operator.