  Since Wilson's work on lattice gauge theory in the 1970s, discrete versions
of field theories have played a vital role in fundamental physics. But there is
recent interest in certain higher dimensional analogues of gauge theory, such
as p-form electromagnetism, including the Kalb-Ramond field in string theory,
and its nonabelian generalizations. It is desirable to discretize such `higher
gauge theories' in a way analogous to lattice gauge theory, but with the
fundamental geometric structures in the discretization boosted in dimension. As
a step toward studying discrete versions of more general higher gauge theories,
we consider the case of p-form electromagnetism. We show that discrete p-form
electromagnetism admits a simple algebraic description in terms of chain
complexes of abelian groups. Moreover, the model allows discrete spacetimes
with quite general geometry, in contrast to the regular cubical lattices
usually associated with lattice gauge theory. After constructing a suitable
model of discrete spacetime for p-form electromagnetism, we quantize the theory
using the Euclidean path integral formalism. The main result is a description
of p-form electromagnetism as a `chain field theory' -- a theory analogous to
topological quantum field theory, but with chain complexes replacing manifolds.
This, in particular, gives a notion of time evolution from one `spacelike
slice' of discrete spacetime to another.
