  Observables in the quantum field theories of $(D-1)$-form fields, $\ca$, on
$D$-dimensional, compact and orientable manifolds, $M$, are computed.
Computations of the vacuum value of $T_{ab}$ find it to be the metric times a
function of the volume of spacetime, $\Omega(M)$. Part of this function of
$\Omega$ is a finite zero-mode contribution. The correlation functions of
another set of operators give intersection numbers on $M$. Furthermore, a
similar computation for products of Wilson area operators results in a function
of the volumes of the intersections of the submanifolds the operators are
defined on. In addition, scalar field couplings are introduced and potentials
are induced after integrating out the $\ca$ field. Lastly, the thermodynamics
of the pure theories is found to be analogous to the zero-point motion of a
scalar particle. The coupling of a Gaussian scalar field to the $\ca$ field is
found to manifest itself on the free energy at high temperatures and/or small
volumes.
