Integrated Correlators and the AdS/CFT Correspondence in N=2 SYM Theories with Fundamental Flavors: A Matrix Model Approach

Core Concepts

This paper investigates the integrated correlators of moment-map operators in the D theory, an N=2 superconformal gauge theory, using a matrix model approach based on supersymmetric localization. By studying the mass-deformed D theory (D* theory) and analyzing its free energy at strong coupling, the authors aim to constrain the scattering amplitudes of gluons in the holographic dual theory described by Type IIB string theory on AdS5 x S5/Γ.

Abstract

• Bibliographic Information: Bill`o, M., Frau, M., Lerda, A., Pini, A., & Vallarino, P. (2024). Integrated correlators in a N = 2 SYM theory with fundamental flavors: a matrix-model perspective. arXiv preprint arXiv:2407.03509v2.
• Research Objective: This research paper aims to study the integrated correlators of moment-map operators in the D theory, an N=2 superconformal gauge theory, using a matrix model approach based on supersymmetric localization. The authors focus on analyzing the mass-deformed D theory (D* theory) and its free energy at strong coupling to gain insights into the scattering amplitudes of gluons in the holographic dual theory.
• Methodology: The authors employ the "full Lie algebra" approach to analyze the matrix model of the mass-deformed D theory arising from localization. This approach allows them to treat double-trace interactions in the matrix model exactly and obtain explicit results for the partition function and its mass derivatives up to O(1/N^2) corrections. They then study the strong coupling limit of these results, which provides valuable information about the dual open string scattering amplitudes in AdS.
• Key Findings: The paper demonstrates that despite the complex structure of the D theory matrix model, exact results can be obtained in the large-N expansion using the full Lie algebra approach. The authors derive explicit expressions for the quartic mass-derivatives of the free energy in terms of integrals of Bessel functions, valid for all values of the 't Hooft coupling. At strong coupling, these mass-derivatives exhibit a behavior similar to those of the Sp(N) theory, suggesting a common holographic dual type for both theories.
• Main Conclusions: The study successfully applies the full Lie algebra approach to the D theory matrix model, overcoming the challenges posed by its double-trace interactions. The analysis of the mass-deformed theory at strong coupling reveals similarities with the Sp(N) theory, implying a shared holographic dual structure. This finding contributes to a deeper understanding of the AdS/CFT correspondence in N=2 SYM theories with fundamental flavors.
• Significance: This research significantly contributes to the understanding of integrated correlators in N=2 superconformal gauge theories and their connection to scattering amplitudes in AdS through the AdS/CFT correspondence. The successful application of the full Lie algebra approach to the D theory matrix model opens up possibilities for studying other complex gauge theories with similar structures.
• Limitations and Future Research: The paper focuses on the large-N limit and considers mass-derivatives up to quartic order. Exploring higher-order corrections and finite-N effects could provide a more complete picture. Additionally, investigating other observables beyond the free energy and extending the analysis to different N=2 SYM theories would be valuable avenues for future research.

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Integrated correlators in a $\mathcal{N}=2$ SYM theory with fundamental flavors: a matrix-model perspective

by M. Billo, M.... at arxiv.org 11-08-2024

https://arxiv.org/pdf/2407.03509.pdf

Deeper Inquiries

How do the findings of this research contribute to our understanding of quantum gravity and string theory through the AdS/CFT correspondence?

This research significantly contributes to our understanding of quantum gravity and string theory through the AdS/CFT correspondence by providing a concrete computational framework to study scattering amplitudes in string theory using the dual gauge theory.
Here's how:

Connecting integrated correlators to scattering amplitudes: The research establishes a direct link between integrated correlators of specific operators (moment-map operators) in the D theory and scattering amplitudes of gluons in the dual AdS space. This connection arises from the AdS/CFT dictionary, where these operators correspond to the gluons in the string theory picture.
Exact results at strong coupling: By employing matrix model techniques and the full Lie algebra approach, the authors calculate the integrated correlators in the D theory exactly in the large-N limit and at strong coupling. This is a highly non-trivial achievement, as strong coupling dynamics in gauge theories are generally difficult to access.
Constraining string theory amplitudes: The exact results for the integrated correlators translate into concrete constraints on the form of the dual gluon scattering amplitudes in AdS. This provides a powerful way to study and potentially verify string theory predictions using gauge theory computations.
Similarities between different theories: The observed similarities in the strong coupling behavior of the D theory and the previously studied Sp(N) theory, despite their different microscopic details, hint at universal features of holographic dualities. This suggests that certain properties of string theory in AdS might be captured by a broader class of gauge theories.
Overall, this research exemplifies the power of the AdS/CFT correspondence in providing a concrete and quantitative link between quantum gravity and gauge theories. The computational tools and insights gained from studying integrated correlators in the D theory pave the way for further exploration of string theory and quantum gravity using dual gauge theory methods.

Could the similarities in the strong coupling behavior of the D theory and Sp(N) theory be attributed to some underlying symmetry or duality that connects these two theories?

The similarities in the strong coupling behavior of the D theory and the Sp(N) theory, despite their different gauge groups and matter content, strongly suggest the existence of an underlying connection, possibly through a symmetry or duality. While a definitive answer requires further investigation, here are some plausible explanations:

Common holographic dual: Both theories are believed to be holographically dual to string theory on AdS5 × S5/Γ, where Γ represents different discrete groups in each case. The D theory corresponds to a combination of orbifold and orientifold projections, while the Sp(N) theory arises from an orientifold projection. The shared AdS factor in their dual geometries could lead to similar strong coupling dynamics, as the AdS/CFT correspondence dictates that the large-N, strong coupling limit of the gauge theory corresponds to the classical limit of the string theory.
Emergent symmetry: It's possible that an underlying symmetry emerges in the strong coupling limit of both theories, even if it's not manifest in their weak coupling descriptions. This emergent symmetry could relate the two theories and explain the observed similarities. Such emergent symmetries are not uncommon in holographic dualities, where the symmetry group on the gravity side can be larger than the apparent symmetry group of the dual gauge theory.
Duality web: The D theory and the Sp(N) theory might be connected through a web of dualities, similar to the intricate web of dualities observed in string theory and supersymmetric gauge theories. This web could involve intermediate dualities with other theories, ultimately linking the D theory and the Sp(N) theory and explaining their shared strong coupling behavior.
Unraveling the precise nature of the connection between the D theory and the Sp(N) theory is an exciting avenue for future research. It could provide valuable insights into the non-perturbative aspects of both gauge theories and string theory, potentially revealing hidden structures and dualities.

How can the matrix model techniques and insights from this study be applied to investigate other aspects of N=2 SYM theories, such as their spectrum of operators, correlation functions, and non-perturbative effects?

The matrix model techniques and insights from this study offer a powerful toolkit to investigate various aspects of N=2 SYM theories beyond integrated correlators. Here are some potential applications:

Spectrum of operators: The matrix model provides a direct way to study the spectrum of protected operators in N=2 SYM theories. By analyzing the eigenvalues and eigenvectors of the matrix model, one can extract information about the scaling dimensions and other quantum numbers of these operators. This approach has been successfully applied to study the spectrum of chiral operators in various N=2 theories.
Correlation functions: While this study focused on integrated correlators, the matrix model techniques can be extended to compute more general correlation functions of local operators. This involves studying higher-point functions within the matrix model and carefully taking into account the operator mixing that can occur at finite N.
Non-perturbative effects: The matrix model can capture certain non-perturbative effects in N=2 SYM theories, such as instantons. These effects manifest as non-trivial contributions to the matrix model path integral, which can be analyzed using techniques like saddle-point approximation or numerical methods.
Deformations and flows: The matrix model framework is well-suited to study deformations of N=2 SYM theories, such as mass deformations or the introduction of Omega background. By incorporating these deformations into the matrix model, one can investigate the resulting renormalization group flows and analyze the low-energy effective theories.
Generalizations to other theories: The full Lie algebra approach and the computational techniques developed in this study can be adapted to study matrix models associated with other N=2 SYM theories, including quiver gauge theories and theories with different matter content. This opens up possibilities to explore a wider class of holographic dualities and gain further insights into the AdS/CFT correspondence.
By leveraging the power of matrix models and the insights gained from this research, we can significantly advance our understanding of N=2 SYM theories, their non-perturbative dynamics, and their connection to string theory and quantum gravity.