Sphere partition functions and the Zamolodchikov metric

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 un-physical. 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.