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Large-N correlation functions in ${\cal N} = 2$ superconformal QCD

Marco Baggio, Vasilis Niarchos, Kyriakos Papadodimas, Gideon Vos

TL;DR

This work computes extremal 2- and 3-point functions of chiral primaries in large-$N$ ${ m SU}(N)$ ${ m obreak N}=2$ SCQCD by leveraging supersymmetric localization to relate correlators to derivatives of the deformed $S^4$ partition function and its planar matrix-model free energy $F_0$. It develops both a basis-dependent and a basis-independent framework, including a tt$^*$-driven recursion for scalars on the conformal manifold, and analyzes weak- and strong-coupling regimes via perturbation theory and the Wiener-Hopf method, respectively. The paper provides explicit tree-level and first subleading weak-coupling results for single-trace correlators, clarifies mixing with multi-trace operators, and introduces parallel-transport and basis-independent observables that remain well-defined along the conformal manifold. Partial strong-coupling results reveal the expected $( ext{log} lambda)^p$ scalings and connect to density-density analyses of the matrix model, offering insights into AdS/CFT duals and the geometry of the conformal manifold in 4d ${ m obreak N}=2$ SCFTs.

Abstract

We study extremal correlation functions of chiral primary operators in the large-N SU(N) ${\cal N} = 2$ superconformal QCD theory and present new results based on supersymmetric localization. We discuss extensively the basis-independent data that can be extracted from these correlators using the leading order large-N matrix model free energy given by the four-sphere partition function. Special emphasis is given to single-trace 2- and 3-point functions as well as a new class of observables that are scalars on the conformal manifold. These new observables are particular quadratic combinations of the structure constants of the chiral ring. At weak 't Hooft coupling we present perturbative results that, in principle, can be extended to arbitrarily high order. We obtain closed-form expressions up to the first subleading order. At strong coupling we provide analogous results based on an approximate Wiener-Hopf method.

Large-N correlation functions in ${\cal N} = 2$ superconformal QCD

TL;DR

This work computes extremal 2- and 3-point functions of chiral primaries in large- SCQCD by leveraging supersymmetric localization to relate correlators to derivatives of the deformed partition function and its planar matrix-model free energy . It develops both a basis-dependent and a basis-independent framework, including a tt-driven recursion for scalars on the conformal manifold, and analyzes weak- and strong-coupling regimes via perturbation theory and the Wiener-Hopf method, respectively. The paper provides explicit tree-level and first subleading weak-coupling results for single-trace correlators, clarifies mixing with multi-trace operators, and introduces parallel-transport and basis-independent observables that remain well-defined along the conformal manifold. Partial strong-coupling results reveal the expected scalings and connect to density-density analyses of the matrix model, offering insights into AdS/CFT duals and the geometry of the conformal manifold in 4d SCFTs.

Abstract

We study extremal correlation functions of chiral primary operators in the large-N SU(N) superconformal QCD theory and present new results based on supersymmetric localization. We discuss extensively the basis-independent data that can be extracted from these correlators using the leading order large-N matrix model free energy given by the four-sphere partition function. Special emphasis is given to single-trace 2- and 3-point functions as well as a new class of observables that are scalars on the conformal manifold. These new observables are particular quadratic combinations of the structure constants of the chiral ring. At weak 't Hooft coupling we present perturbative results that, in principle, can be extended to arbitrarily high order. We obtain closed-form expressions up to the first subleading order. At strong coupling we provide analogous results based on an approximate Wiener-Hopf method.

Paper Structure

This paper contains 30 sections, 200 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Exact 2- and 3-point functions in the ${\cal N}=2$ chiral ring
  3. $SU(N)$ ${\cal N}=2$ chiral ring
  4. Extremal correlation functions in the ${\cal N}=2$ chiral ring
  5. 2- and 3-point functions from $S^4$ partition functions and matrix models
  6. Deformed partition functions on $S^4$ and their localization
  7. Relation between the $S^4$ partition function and 2-point functions on ${\mathbb R}^4$
  8. Formulae for 3-point functions
  9. Correlation functions at large $N$
  10. Correlation functions and the matrix model free energy at large $N$
  11. Single-trace 2- and 3-point functions
  12. Mixing with multi-trace operators
  13. Basis-independent 3-point functions
  14. $tt^*$ equations and $|C^{(2,\Delta)}|^2$
  15. Weak coupling results
  16. ...and 15 more sections

Figures (1)

  • Figure 1: The dotted line represents the numerical solution of $\mu$ as a function of $\log(\lambda)$ after all other source terms have been set to zero, for comparison the solid line represents a plot of $\mu=\frac{2}{\pi}\log(\lambda)$