[Paper Review] Introduction to the AdS/CFT correspondence

Alfonso V. Ramallo|arXiv (Cornell University)|Oct 16, 2013

Black Holes and Theoretical Physics27 references17 citations

TL;DR

This paper provides a pedagogical introduction to the AdS/CFT correspondence, explaining how strongly coupled quantum field theories in d dimensions are dual to classical gravity in (d+1)-dimensional anti-de Sitter (AdS) space. It derives key results such as the holographic calculation of correlation functions, quark-antiquark potentials, and transport coefficients like shear viscosity, showing that the KSS bound η/s ≥ 1/(4π) is saturated in certain gravity models and violated in higher-curvature theories.

ABSTRACT

This is a pedagogical introduction to the AdS/CFT correspondence, based on lectures delivered by the author at the third IDPASC school. Starting with the conceptual basis of the holographic dualities, the subject is developed emphasizing some concrete topics, which are discussed in detail. A very brief introduction to string theory is provided, containing the minimal ingredients to understand the origin of the AdS/CFT duality. Other topics covered are the holographic calculation of correlation functions, quark-antiquark potentials and transport coefficients.

Motivation & Objective

• To provide a conceptual and calculational introduction to the AdS/CFT correspondence for researchers new to the field.
• To explain how the renormalization group flow in quantum field theories can be geometrically realized as a gravity theory in higher dimensions.
• To demonstrate practical applications of the duality, including correlation functions, quark-antiquark potentials, and transport coefficients.
• To connect the duality to condensed matter physics, quantum gravity, and hydrodynamics, highlighting its broad applicability.
• To lay the foundation for extending the duality to non-supersymmetric and confining theories, and to include fundamental matter fields via flavor branes.

Proposed method

• Uses the Kadanoff-Wilson renormalization group framework to motivate the emergence of an extra dimension in the dual gravity description.
• Models the running couplings of a quantum field theory as bulk fields in a higher-dimensional AdS space, with sources identified as boundary values of these fields.
• Applies linear response theory and correlation function calculations in AdS space to compute two-point functions and one-point functions.
• Computes the quark-antiquark potential in both finite-temperature and confining backgrounds using the Nambu-Goto action for a string in curved spacetime.
• Analyzes black hole thermodynamics in AdS to relate the entropy and temperature of black holes to the thermal properties of the dual field theory.
• Derives the shear viscosity-to-entropy density ratio η/s in Einstein gravity and Gauss-Bonnet gravity, showing the KSS bound and its violation.

Experimental results

Research questions

• RQ1 How can the renormalization group flow of a strongly coupled quantum field theory be geometrically realized as a gravity theory in one higher dimension?
• RQ2 What is the holographic relation between correlation functions in a conformal field theory and bulk field propagation in AdS space?
• RQ3 How is the quark-antiquark potential computed in the AdS/CFT framework, and what does it reveal about screening and confinement?
• RQ4 What is the origin of the KSS bound on the shear viscosity-to-entropy ratio, and how is it modified in higher-curvature gravity theories?
• RQ5 How can the AdS/CFT correspondence be extended to describe non-supersymmetric, confining, or flavored field theories?

Key findings

• The AdS/CFT correspondence realizes the renormalization group flow as a classical gravity theory in a higher-dimensional AdS space, with the extra dimension corresponding to the energy scale.
• Correlation functions in the boundary CFT are computed via the on-shell action of bulk fields, with two-point functions derived from the quadratic action of a scalar field in AdS.
• The quark-antiquark potential in a finite-temperature plasma is computed using a string worldsheet in the AdS-Schwarzschild background, showing screening at high temperatures.
• In the confining background, the potential exhibits linear confinement, with the string stretching between the quark and antiquark ending on a horizon.
• The shear viscosity-to-entropy ratio in Einstein gravity is η/s = 1/(4π), saturating the KSS bound, while in Gauss-Bonnet gravity it becomes η/s = (1 - 4λ_GB)/(4π), violating the bound for λ_GB > 0.
• The duality allows for the holographic description of strongly correlated systems in condensed matter, such as strange metals and superconductors, via gravity models with appropriate boundary conditions.