Kähler potential and ambiguities in 4d NN = 2 SCFTs

The partition function of four-dimensional N = 2 superconformal field theories on S4 computes the exact Kähler potential on the space of exactly marginal couplings [1]. We present a new elementary proof of this result using supersymmetry Ward identities. The partition function is a section rather than a function, and is subject to ambiguities coming from Kähler transformations acting on the Kähler potential. This ambiguity is realized by a local supergravity counterterm in the underlying SCFT. We provide an explicit construction of the Kähler ambiguity counterterm in the four dimensional N = 2 off-shell supergravity theory that admits S4 as a supersymmetric background.