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 unphysical. However, in the presence of additional symmetries, the partition function of even-dimensional CFTs can become physical. For two-dimensional (formula presented) supersymmetric CFTs, the continuum partition function exists and computes the Kähler 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 Kähler transformation ambiguity is identified with a local term in the corresponding (formula presented) supergravity theory. We derive an analogous, new, result in the case of four-dimensional (formula presented) supersymmetric CFTs: the S⁴ partition function computes the Kähler potential on the superconformal manifold. Finally, we show that (formula presented) supersymmetry in four dimensions and (formula presented) supersymmetry in two dimensions are not sufficient to make the corresponding sphere partition functions well-defined functions of the exactly marginal parameters.