[Paper Review] Introduction to Gauge/Gravity Duality (TASI Lectures 2017)

Johanna Erdmenger|arXiv (Cornell University)|Jul 25, 2018

Black Holes and Theoretical Physics42 references3 citations

TL;DR

This paper introduces gauge/gravity duality through the AdS/CFT correspondence, emphasizing its origin in string theory and generalization to strongly coupled systems. It presents a holographic Kondo model where a spin impurity coupled to a strongly coupled electron gas is dual to a gravity background with a black hole, reproducing key condensed matter phenomena like logarithmic resistivity and power-law scaling, with spectral asymmetry linked to black hole entropy in AdS₂.

ABSTRACT

We review how the AdS/CFT correspondence is motivated within string theory, and discuss how it is generalized to gauge/gravity duality. In particular, we highlight the relation to quantum information theory by pointing out that the Fisher information metric of a Gaussian probability distribution corresponds to an Anti-de Sitter space. As an application example of gauge/gravity duality, we present a holographic Kondo model. The Kondo model in condensed matter physics describes a spin impurity interacting with a free electron gas: At low energies, the impurity is screened and there is a logarithmic rise of the resistivity. In quantum field theory, this amounts to a negative beta function for the impurity coupling and the theory flows to a non-trivial IR fixed point. For constructing a gravity dual, we consider a large $N$ version of this model in which the ambient electrons are strongly coupled even before the interaction with the impurity is switched on. We present the brane construction which motivates a gravity dual Kondo model and use this model to calculate the impurity entanglement entropy and the resistivity, which has a power-law behaviour. We also study quantum quenches, and discuss the relation to the Sachdev-Ye-Kitaev model.

Motivation & Objective

• To motivate gauge/gravity duality via the AdS/CFT correspondence within string theory and its generalization to non-conformal, strongly coupled systems.
• To establish a gravity dual for the large-N Kondo model in condensed matter physics, capturing non-Fermi liquid behavior and screening of spin impurities.
• To connect the holographic Kondo model to quantum information via the Fisher information metric and black hole entropy in AdS₂.
• To explore quantum quenches and correlation functions in the holographic Kondo model, linking to the Sachdev-Ye-Kitaev model.
• To demonstrate how holography enables computation of observables in strongly coupled field theories otherwise intractable via standard methods.

Proposed method

• Use the AdS/CFT correspondence in the large-N, strong coupling limit to map a strongly coupled gauge theory with a spin impurity to a gravity dual with a black hole and brane construction.
• Construct the gravity dual using a D3-brane and D7-brane system to realize the Kondo effect, with the impurity coupling via a double-trace deformation.
• Compute the retarded Green's function and extract the spectral function, showing a pole at ωP ∝ −i|⟨O⟩|², indicating spectral asymmetry.
• Calculate the impurity entanglement entropy and resistivity, finding power-law behavior consistent with the Kondo effect.
• Analyze quantum quenches in the holographic model, comparing dynamics to the Sachdev-Ye-Kitaev model.
• Map the spectral asymmetry in the Kondo model to black hole entropy in two-dimensional AdS space, drawing parallels with the SYK model.

Experimental results

Research questions

• RQ1 How can the Kondo effect in condensed matter physics be realized in a gravity dual framework via gauge/gravity duality?
• RQ2 What is the role of the Fisher information metric in connecting Gaussian probability distributions to Anti-de Sitter geometry?
• RQ3 How does the holographic Kondo model reproduce the logarithmic resistivity rise and non-Fermi liquid behavior of the large-N Kondo model?
• RQ4 What is the origin of spectral asymmetry in the retarded Green's function, and how is it related to black hole entropy in AdS₂?
• RQ5 How do quantum quench dynamics in the holographic Kondo model compare to those in the Sachdev-Ye-Kitaev model?

Key findings

• The holographic Kondo model reproduces the power-law resistivity behavior characteristic of the Kondo effect, confirming the non-Fermi liquid nature of the system.
• The impurity entanglement entropy is computed via the Ryu-Takayanagi formula in the gravity dual, yielding a non-trivial scaling with temperature.
• The retarded Green's function exhibits a pole at ωP ∝ −i|⟨O⟩|², confirming spectral asymmetry consistent with the Kondo screening mechanism.
• The spectral asymmetry in the Kondo model is mapped to black hole entropy in AdS₂, mirroring the correspondence found in the Sachdev-Ye-Kitaev model.
• Quantum quench dynamics in the holographic Kondo model show thermalization and non-equilibrium behavior consistent with strongly correlated systems.
• The model realizes a non-trivial infrared fixed point with a negative beta function for the impurity coupling, confirming the theory's flow to strong coupling.