We compute exact 2-and 3-point functions of chiral primaries in fourdimensional 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 Kähler 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.
We consider the world-sheet S matrix of superstrings on an AdS 3 × S 3 × T 4 NS-NS background in uniform light-cone gauge. We argue that scattering is given by a CDD factor that is nontrivial only between opposite-chirality particles, yielding a spin-chain-like Bethe ansatz. Our proposal reproduces the spectrum of nonprotected states computed from the Wess-Zumino-Witten description and the perturbative tree-level S matrix. This suggests that the model is an integrable deformation of a free theory similar to those arising from the TT composite operator.
In this note we prove a non-renormalization theorem for the 3-point functions of 1/2 BPS primaries in the four-dimensional N = 4 SYM and chiral primaries in two dimensional N = (4, 4) SCFTs. Our proof is rather elementary: it is based on Ward identities and the structure of the short multiplets of the superconformal algebra and it does not rely on superspace techniques. We also discuss some possible generalizations to less supersymmetric multiplets.
We consider the exact coupling constant dependence of extremal correlation functions of N = 2 chiral primary operators in 4d N = 2 superconformal gauge theories with gauge group SU (N ) and N f = 2N massless fundamental hypermultiplets. The 2-and 3-point functions, viewed as functions of the exactly marginal coupling constant and theta angle, obey the tt * equations. In the case at hand, the tt * equations form a set of complicated non-linear coupled matrix equations. We point out that there is an ad hoc self-consistent ansatz that reduces this set of partial differential equations to a sequence of decoupled semi-infinite Toda chains, similar to the one encountered previously in the special case of SU (2) gauge group. This ansatz requires a surprising new non-renormalization theorem in N = 2 superconformal field theories. We derive a general 3-loop perturbative formula for 2-and 3-point functions in the N = 2 chiral ring of the SU (N ) theory, and in all explicitly computed examples we find agreement with the tt * equations, as well as the above-mentioned ansatz. This is suggestive evidence for an interesting non-perturbative conjecture about the structure of the N = 2 chiral ring in this class of theories. We discuss several implications of this conjecture. For example, it implies that the holonomy of the vector bundles of chiral primaries over the superconformal manifold is reducible. It also implies that a specific subset of extremal correlation functions can be computed in the SU (N ) theory using information solely from the S 4 partition function of the theory obtained by supersymmetric localization.
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