AdS$_2$ holography and the SYK model

TL;DR

This work surveys AdS2 holography and the SYK model, focusing on how near-extremal black holes exhibit universal infrared dynamics captured by Jackiw–Teitelboim gravity and its Schwarzian boundary action. It details how backreaction and dilaton gravity constrain the AdS2 region, motivating a cutoff AdS2 description that yields a universal IR theory and a concrete holographic dictionary. The SYK model is analyzed at large N, showing emergent reparametrization symmetry, a conformal IR limit with Δ=1/q, and a tunable Schwarzian sector that mirrors JT gravity, culminating in a calculable four-point function and chaotic behavior with maximal Lyapunov exponent; discussions on a bulk dual emphasize a nonlocal or kinematic-space viewpoint with an infinite bulk mode tower. Overall, the notes highlight the deep connections between AdS2 holography, JT gravity, Schwarzian dynamics, and SYK-like quantum chaotic systems, while outlining open questions about a precise bulk dual and UV completions.

Abstract

These are lecture notes based on a series of lectures presented at the XIII Modave Summer School in Mathematical physics aimed at PhD students and young postdocs. The goal is to give an introduction to some of the recent developments in understanding holography in two bulk dimensions, and its connection to microscopics of near extremal black holes. The first part reviews the motivation to study, and the problems (and their interpretations) with holography for AdS$_2$ spaces. The second part is about the Jackiw-Teitelboim theory and nearly-AdS$_2$ spaces. The third part introduces the Sachdev-Ye-Kitaev model, reviews some of the basic calculations and discusses what features make the model exciting.

Figures (8)

• Figure 1: Left: Maximally extended Penrose diagram of the extremal Reisner-Nordström solution. The blue region zigg-zagging up is the AdS$_2$ near-horizon region. The red dashed region is the patch covered by Poincaré coordinates. Right: Penrose diagram and coordinates of global AdS$_2$. Notice that there are two boundaries. The Poincaré patch is the light yellow region, while the dark yellow region is the Rindler patch.
• Figure 2: Left: Coordinates on the hyperbolic disk. Right: A cutout from the hyperbolic disk.
• Figure 3: Left: Original contour. Right: Deformed contour.
• Figure 4: Left: Divergence of trajectories in phase space. Right: Typical behaviour of \ref{['eq:commsq']}.
• Figure 5: Left: Histogram of the spectrum of the $q=2$ model. Right: Histogram of the spectrum of the $q=4$ model. Both plots are for $N=20$ and can be produced with the use of the recursion relations \ref{['eq:recursion']} on a laptop while drinking a smaller cup of coffee.