  We study the finite part of the sphere partition function of d-dimensional
Conformal Field Theories (CFTs) as a function of exactly marginal couplings. In
odd dimensions, this quantity is physical and independent of the exactly
marginal couplings. In even dimensions, this object is generally regularization
scheme dependent and thus unphysical. However, in the presence of additional
symmetries, the partition function of even-dimensional CFTs can become
physical. For two-dimensional N=(2,2) supersymmetric CFTs, the continuum
partition function exists and computes the Kahler potential on the chiral and
twisted chiral superconformal manifolds. We provide a new elementary proof of
this result using Ward identities on the sphere. The Kahler transformation
ambiguity is identified with a local term in the corresponding N=(2,2)
supergravity theory. We derive an analogous, new, result in the case of
four-dimensional N=2 supersymmetric CFTs: the S^4 partition function computes
the Kahler potential on the superconformal manifold. Finally, we show that N=1
supersymmetry in four dimensions and N=(1,1) supersymmetry in two dimensions
are not sufficient to make the corresponding sphere partition functions
well-defined functions of the exactly marginal parameters.
