Correlation Functions of Coulomb Branch Operators (1602.05971v2)

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.

Overview of Extremal Correlators in 4D $\mathcal{N}=2$ Superconformal Field Theories

This paper explores the intricate computation of extremal correlators in four-dimensional $\mathcal{N}=2$ Superconformal Field Theories (SCFTs), focusing on correlators that involve a multitude of chiral primary operators alongside a singular anti-chiral primary operator. These correlators are distinctly termed "extremal" due to their nature as non-holomorphic observables achieved via minimal configuration in the theory.

The authors establish a significant connection between extremal correlators and the partition function on a deformed four-sphere ($S^4$). This non-trivial relation emerges from handling the challenge posed by conformal anomalies which provoke operator mixing on the sphere. The methodology primarily leverages supersymmetric localization to compute the deformed $S^4$ partition functions, assessing extremal correlators in substantial examples across different SCFTs.

Key Findings and Methodology

1. Correlation Function Calculation:
   • The extremal correlators, consisting of $(n-1)$ chiral operators and one anti-chiral operator, are expressed through determinants of derivatives of the $S^4$ partition function.
   • This involves calculating explicitly many examples through supersymmetric localization.
2. Integrable Differential Equations:
   • A notable contribution to theory comes in the form of establishing that extremal correlators are governed by a system of integrable differential equations, which provides a mathematical framework to systematize these computations.
3. Extension of Previous Formulations:
   • The research builds upon earlier works, notably expanding upon formulas connecting undeformed partition functions and Kähler potential on SCFT conformal manifolds.
   • A generalized algorithm for the computation of these correlators under various $\mathcal{N}=2$ theories is outlined.
4. Exact Results and Perturbative Comparisons:
   • Comparison of obtained results with perturbative computations and with four-dimensional $tt^*$ equations reveals both congruence and novelty in the approach.

Practical and Theoretical Implications

• Practical:
  • The capability to compute these extremal correlators exactly could benefit numerous applications in theoretical and computational physics, particularly in areas exploring quantum field dynamics and supersymmetry.
• Theoretical:
  • Establishing an integrable system for extremal correlators enriches understanding and potential applications in determining spectral data of SCFTs.
  • The paper unveils the algebraic structure by way of connecting computational models to moduli space physics, aiding further exploration in SCFT conformal manifold analysis.

Future Directions

The current work opens avenues for dissecting duality transformations concerning extremal correlators, potentially offering deeper insights into how various points within a conformal manifold might transform under strong-weak coupling scenarios. The authors suggest extensions into exploring these correlators with added supersymmetric defects or external operators such as Wilson loops, providing comprehensive data on the dynamic landscape of SCFTs.

Conclusion

The paper is heavy with both theoretical rigor and computational precision in handling the deformed partition function approach to extremal correlators, bolstered by an overview of mathematical models and algebraic geometry. It promises to serve as a cornerstone for developing further computational techniques in quantum field theory and offers significant theoretical insights into numerically analyzing higher-order symmetries intrinsic to four-dimensional $\mathcal{N}=2$ SCFTs.