Large N Correlation Functions in Superconformal Field Theories

TL;DR

This work computes exact two-point (and related) correlators of chiral primary operators in large-$N$ ${\cal N}=4$ SYM and ${\cal N}=2$ SCQCD by leveraging supersymmetric localization and derivatives of the $S^4$ partition function with respect to chiral-ring couplings. After accounting for conformal-anomaly–induced mixing on $S^4$ via Gram-Schmidt, the authors connect sphere data to flat-space correlators through a large-$N$ saddle-point analysis of a deformed matrix model, focusing on single-trace operators ${\rm Tr}\phi^n$. For ${\cal N}=4$, the resulting correlators reproduce free-field values due to a non-renormalization theorem, while in SCQCD the leading planar result matches the free theory and the authors compute explicit weak-coupling corrections (including $\zeta(3)$ terms) that admit a finite-$N$ extension consistent with known small-$N$ results. The work also discusses strong-coupling expectations and outlines avenues for higher-point functions and nonperturbative corrections. Overall, the paper provides a framework to obtain exact large-$N$ CPO correlators from localization data, bridging sphere partition functions, conformal-manifold geometry, and flat-space observables with potential for broad applications.

Abstract

We compute correlation functions of chiral primary operators in N=2 superconformal theories at large N using a construction based on supersymmetric localization recently developed by Gerchkovitz et al. We focus on N=4 SYM as well as on superconformal QCD. In the case of N=4 we recover the free field theory results as expected due to non-renormalization theorems. In the case of superconformal QCD we study the planar expansion in the large N limit. The final correlators admit a simple generalization to a finite N formula which exactly matches the various small $N$ results in the literature.