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We study the free energy of four-dimensional CFTs on deformed spheres. Forgeneric nonsupersymmetric CFTs only the coefficient of the logarithmicdivergence in the free energy is physical, which is an extremum for the roundsphere. We then specialize to $\mathcal{N}=2$ SCFTs where one can preserve somesupersymmetry on a compact manifold by turning on appropriate backgroundfields. For deformations of the round sphere the $c$ anomaly receivescorrections proportional to the supersymmetric completion of the (Weyl)$^2$term, which we determine up to one constant by analyzing the scale dependenceof various correlators in the stress-tensor multiplet. We further show that thedouble derivative of the free energy with respect to the marginal couplings isproportional to the two-point function of the bottom components of the marginalchiral multiplet placed at the two poles of the deformed sphere. We then useanomaly considerations and counter-terms to parametrize the finite part of thefree energy which makes manifest its dependence on the K\"ahler potential. Wedemonstrate these results for a theory with a vector multiplet and a masslessadjoint hypermultiplet using results from localization. Finally, by choosing aspecial value of the hypermultiplet mass where the free energy is independentof the deformation, we derive an infinite number of constraints between variousintegrated correlators in $\mathcal{N}=4$ super Yang-Mills with any gauge groupand at all values of the coupling, extending previous results.

Conformal field theories on deformed spheres, anomalies, and supersymmetry