  We show that the determinant of the co-exact $p$-form on spheres and
anti-deSitter spaces can be written as an integral transform of bulk and edge
Harish-Chandra characters. The edge character of a co-exact $p$-form contains
characters of anti-symmetric tensors of rank lower to $p$ all the way to the
zero-form. Using this result we evaluate the partition function of $p$-forms
and demonstrate that they obey known properties under Hodge duality. We show
that partition function of conformal forms in even $d+1$ dimensions, on
hyperbolic cylinders can be written as integral transforms involving only the
bulk characters. This supports earlier observations that entanglement entropy
evaluated using partition functions on hyperbolic cylinders do not contain
contributions from the edge modes. For conformal coupled scalars we demonstrate
that the character integral representation of the free energy on hyperbolic
cylinders and branched spheres coincide. Finally we propose a character
integral representation for the partition function of $p$-forms on branched
spheres.
