tt$^{*}$ equations, localization and exact chiral rings in 4d $ \mathcal{N} $ =2 SCFTs

We compute exact 2- and 3-point functions of chiral primaries in four-dimensional N=2 superconformal field theories, including all perturbative and instanton contributions. We demonstrate that these correlation functions are nontrivial and satisfy exact differential equations with respect to the coupling constants. These equations are the analogue of the $tt^*$ equations in two dimensions. In the SU(2) N=2 SYM theory coupled to 4 hypermultiplets they take the form of a semi-infinite Toda chain. We provide the complete solution of this chain using input from supersymmetric localization. To test our results we calculate the same correlation functions independently using Feynman diagrams up to 2-loops and we find perfect agreement up to the relevant order. As a spin-off, we perform a 2-loop check of the recent proposal of arXiv:1405.7271 that the logarithm of the sphere partition function in N=2 SCFTs determines the K\"ahler potential of the Zamolodchikov metric on the conformal manifold. We also present the $tt^*$ equations in general SU(N) N=2 superconformal QCD theories and comment on their structure and implications.