Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Correlation Functions of Coulomb Branch Operators

Efrat Gerchkovitz, Jaume Gomis, Nafiz Ishtiaque, Avner Karasik, Zohar Komargodski, Silviu S. Pufu

TL;DR

This work develops an exact framework for computing extremal correlators in 4d ${ m N}=2$ SCFTs with a single anti-chiral insertion by linking flat-space correlators to derivatives of a deformed $S^4$ partition function $Z[S^4]$ computed via localization. Operator mixing from conformal anomalies on $S^4$ is addressed by a Gram-Schmidt diagonalization, yielding determinant formulas that relate chiral-ring data to the deformed partition function and satisfy integrable Toda equations in certain cases. The authors apply the method to multiple theories, including ${ m SU}(2)$ SQCD with four fundamentals and ${ m N}=4$ limits, obtaining perturbative and instanton corrections and revealing when decoupled Toda chains persist (up to two loops) and when they break (at higher loops). The results connect localization, conformal-manifold geometry via the Zamolodchikov metric, and integrable structures (Toda, Hitchin systems, ${tt}^*$) and offer new, exact insights into asymptotic perturbative expansions and duality properties of 4d ${ m N}=2$ theories.

Abstract

We consider the correlation functions of Coulomb branch operators in four-dimensional N=2 Superconformal Field Theories (SCFTs) involving exactly one anti-chiral operator. These extremal correlators are the "minimal" non-holomorphic local observables in the theory. We show that they can be expressed in terms of certain determinants of derivatives of the four-sphere partition function of an appropriate deformation of the SCFT. This relation between the extremal correlators and the deformed four-sphere partition function is non-trivial due to the presence of conformal anomalies, which lead to operator mixing on the sphere. Evaluating the deformed four-sphere partition function using supersymmetric localization, we compute the extremal correlators explicitly in many interesting examples. Additionally, the representation of the extremal correlators mentioned above leads to a system of integrable differential equations. We compare our exact results with previous perturbative computations and with the four-dimensional tt^* equations. We also use our results to study some of the asymptotic properties of the perturbative series expansions we obtain in N=2 SQCD.

Correlation Functions of Coulomb Branch Operators

TL;DR

This work develops an exact framework for computing extremal correlators in 4d SCFTs with a single anti-chiral insertion by linking flat-space correlators to derivatives of a deformed partition function computed via localization. Operator mixing from conformal anomalies on is addressed by a Gram-Schmidt diagonalization, yielding determinant formulas that relate chiral-ring data to the deformed partition function and satisfy integrable Toda equations in certain cases. The authors apply the method to multiple theories, including SQCD with four fundamentals and limits, obtaining perturbative and instanton corrections and revealing when decoupled Toda chains persist (up to two loops) and when they break (at higher loops). The results connect localization, conformal-manifold geometry via the Zamolodchikov metric, and integrable structures (Toda, Hitchin systems, ) and offer new, exact insights into asymptotic perturbative expansions and duality properties of 4d theories.

Abstract

We consider the correlation functions of Coulomb branch operators in four-dimensional N=2 Superconformal Field Theories (SCFTs) involving exactly one anti-chiral operator. These extremal correlators are the "minimal" non-holomorphic local observables in the theory. We show that they can be expressed in terms of certain determinants of derivatives of the four-sphere partition function of an appropriate deformation of the SCFT. This relation between the extremal correlators and the deformed four-sphere partition function is non-trivial due to the presence of conformal anomalies, which lead to operator mixing on the sphere. Evaluating the deformed four-sphere partition function using supersymmetric localization, we compute the extremal correlators explicitly in many interesting examples. Additionally, the representation of the extremal correlators mentioned above leads to a system of integrable differential equations. We compare our exact results with previous perturbative computations and with the four-dimensional tt^* equations. We also use our results to study some of the asymptotic properties of the perturbative series expansions we obtain in N=2 SQCD.

Paper Structure

This paper contains 25 sections, 145 equations, 4 figures.

Table of Contents

  1. Introduction and Conclusions
  2. Conformal Manifolds
  3. The Chiral Ring of $\mathcal{N}=2$ SCFTs
  4. Subtle Aspects of Conformal Field Theories on $S^4$
  5. The Chiral Ring in 4d ${\mathcal{N}}=2$ SCFTs and $S^4$
  6. Placing the Deformed Theory on $S^4$
  7. Chiral Primary Correlators from the Deformed Partition Function
  8. The Deformed Partition Function on $S^4$
  9. The Relation Between Correlators in $\mathbb{R}^4$ and $S^4$
  10. Summary of the Algorithm
  11. Examples
  12. $SU(2)$ Gauge Group
  13. $SU(2)$ with an Adjoint Hypermultiplet
  14. $SU(2)$ SQCD with Four Fundamental Hypermultiplets
  15. $SU(N)$ Gauge Group
  16. ...and 10 more sections

Figures (4)

  • Figure 1: The ratio of consecutive coefficients appearing in the perturbative expansion \ref{['G2Expansion']} of $G_2$ in $SU(2)$ SQCD plotted in terms of the loop order $n$.
  • Figure 2: The relative difference between the Padé estimate of the coefficient $a_{n+1}$ and its actual value in the case of $G_2$ in $SU(2)$ SQCD. The black line is a linear fit for $n \geq 40$.
  • Figure 3: Ratios of consecutive coefficients in the series expansions \ref{['GnExpansion']} in the case of $SU(3)$ SQCD.
  • Figure 4: The relative difference between the Padé estimate of the coefficient $a_{m, n+1}$ and its actual value in the case of $G_2$ and $G_3$ in $SU(3)$ SQCD. The black lines are linear fits for $n \geq 40$.