On exact correlation functions in SU(N) ${\cal N} = 2$ superconformal QCD

TL;DR

This work probes the exact coupling dependence of extremal correlators in 4d ${ m SU}(N)$ ${ m N}=2$ SCQCD with ${N_f}=2N$, focusing on the ${tt}^*$ equations for chiral primaries. It introduces a no-mixing, holomorphic basis based on $C_2$-primaries that decouples the nonlinear matrix differential equations into independent semi-infinite Toda chains, linking the chiral-ring data to the Zamolodchikov metric computed from the ${S^4}$ partition function. The authors provide a nontrivial 3-loop perturbative check in SU$(3)$ and SU$(4)$ that supports the decoupling and propose a recursive, finite-coupling solution for each decoupled subsector, with consequences for the holonomy of chiral-primary bundles and computability of certain correlators from localization data. Perturbative results and large-$N$ considerations yield concrete predictions for single-trace extremal correlators and suggest deep structural constraints on the ${ m N}=2$ chiral ring, pointing toward a broader non-perturbative framework for these theories.

Abstract

We consider the exact coupling constant dependence of extremal correlation functions of ${\cal N} = 2$ chiral primary operators in 4d ${\cal N} = 2$ 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 ${\cal N} = 2$ superconformal field theories. We derive a general 3-loop perturbative formula for 2- and 3-point functions in the ${\cal N} = 2$ 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 ${\cal N} = 2$ 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.