  We consider the entanglement entropy arising from edge-modes in Abelian
$p$-form topological field theories in $d$ dimensions on arbitrary spatial
topology and across arbitrary entangling surfaces. We find a series of
descending area laws plus universal corrections proportional to the Betti
numbers of the entangling surface, which can be taken as a higher-dimensional
version of the "topological entanglement entropy." Our calculation comes in two
flavors: firstly, through an induced edge-mode theory appearing on the
regulated entangling surface in a replica path integral and secondly through a
more rigorous definition of the entanglement entropy through an extended
Hilbert space. Along the way we establish several key results that are of their
own merit. We explain how the edge-mode theory is a novel combination of
$(p-1)$-form and $(d-p-2)$-form Maxwell theories linked by a chirality
condition, in what we coin a "chiral mixed Maxwell theory." We explicitly
evaluate the thermal partition function of this theory. Additionally we show
that the extended Hilbert space is completely organized into representations of
an infinite-dimensional, centrally extended current algebra which naturally
generalizes 2d Kac-Moody algebras to arbitrary dimension and topology. We
construct the Verma modules and the representation characters of this algebra.
Lastly, we connect the two approaches, showing that the thermal partition
function of the chiral mixed Maxwell theory is precisely an extended
representation character of our current algebra, establishing an exact
correspondence of the edge-mode theory and the entanglement spectrum.
