Abstract
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.
Highlights
IntroductionEquations of [4], the exact Zamolodchikov metric is a very useful datum that leads to exact information about more general properties of the chiral ring structure of N = 2 superconformal field theory (SCFT). it provides useful input towards an exact solution of the tt∗ equations, which encodes the non-perturbative dependence of 2- and 3-point functions of chiral primary operators on the marginal couplings of the SCFT
In this paper we are interested in four-dimensional theories with N = 2 superconformal invariance
It is of considerable interest to determine how the physical properties of these theories vary as we change the continuous parameters that parametrize these manifolds
Summary
Equations of [4], the exact Zamolodchikov metric is a very useful datum that leads to exact information about more general properties of the chiral ring structure of N = 2 SCFTs. it provides useful input towards an exact solution of the tt∗ equations, which encodes the non-perturbative dependence of 2- and 3-point functions of chiral primary operators on the marginal couplings of the SCFT. It provides useful input towards an exact solution of the tt∗ equations, which encodes the non-perturbative dependence of 2- and 3-point functions of chiral primary operators on the marginal couplings of the SCFT In this solution, correlation functions of chiral primaries with scaling dimension greater than two are expressed in terms of more than two derivatives of the S4 partition function.
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