Chaos in AdS$_2$ holography (1605.06098v3)

Abstract: We revisit AdS$_2$ holography with the Sachdev-Ye-Kitaev models in mind. Our main result is to rewrite a generic theory of gravity near an AdS$_2$ throat as a novel hydrodynamics coupled to the correlation functions of a conformal quantum mechanics. This gives a prescription for the computation of $n$-point functions in the dual quantum mechanics. We thereby find that the dual is maximally chaotic.

• The paper formulates a hydrodynamic framework that links the SYK model's infrared conformal symmetry to gravity in AdS2, establishing maximal chaos.
• It demonstrates that the SYK model saturates the chaos bound through analysis of Lyapunov exponents in four-point functions.
• It addresses challenges in the AdS2/CFT1 correspondence by unifying duality aspects with hydrodynamic reformulations of emergent gravitational dynamics.

Overview of "Chaos in AdS$_2$ holography"

This paper investigates the AdS$_2$ holography by utilizing the Sachdev-Ye-Kitaev (SYK) model as a reference point. The principal achievement is the formulation of a theory where gravity near an AdS$_2$ throat is illustrated as a novel hydrodynamics married with the correlation functions of a conformal quantum mechanics (CQM). This formulation provides a method to calculate $n$-point functions in the dual quantum mechanics, uncovering that the dual is maximally chaotic.

Key Contributions

1. SYK Models and Emergent Gravity Dual: The SYK model, characterized by random interactions among $2N$ Majorana fermions, emerges as a soluble quantum mechanical system at large $N$. It exhibits an emergent conformal symmetry in the infrared (IR), alongside a large $N$ extremal entropy which lends credence to its conjectured gravitational dual in AdS$_2$.
2. Chaotic Nature of the SYK Model: The paper establishes the saturation of the chaos bound by the SYK model, indicating maximal chaos. This is characterized by the Lyapunov exponent that measures the rate of growth in out-of-time-ordered four-point functions.
3. Duality and CQM Paradox: Challenges pertaining to the AdS$_2$/CFT$_1$ correspondence are revisited, particularly addressing the dual CQM as a system reflecting topological yet non-dynamical nature in isolation, further complicated by the UV/IR mixing.
4. Hydrodynamic Reformulation: The paper derives an effective hydrodynamic action for systems flowing into an AdS$_2$ region. This novel hydrodynamics is indicative of the saturation of the chaos bound and crucially links with the diffeomorphism Ward identity in the boundary quantum mechanics.
5. Four-point Functions and Maximal Chaos: The analysis unfolds how quadratic fluctuations in boundary operators inject energy, affecting the boundary theory's dynamics expressed through a holographic black hole interpretation. This demonstrates non-conformal corrections emerging from hydrodynamic backreaction, integral in computing chaotic behavior manifest as a Lyapunov exponent matching that of systems with Einstein gravity duals.

Implications and Future Directions

Practically, this investigation advances the understanding of low-dimensional quantum gravity through models exhibiting emergent conformal symmetry, specifically in two-dimensional systems inspired by the SYK model. Theoretically, the introduction of hydrodynamics in the context of holographic AdS$_2$ spacetimes offers a robust framework to analyze gravitational dynamics and chaos in lower-dimensional dualities.

Looking towards future exploration, the discussion about a gauged SYK/AdS correspondence or potential corrections to the hydrodynamic theory at more comprehensive orders ($\mathcal{O}(\ell^2)$) opens avenues for refining or broadening the scope of these insights. Additionally, scrutinizing the Schwarzian action's universality or studying higher-order quantum corrections might further illuminate their role in large $N$ systems with emergent conformal invariance.

Overall, this work strengthens the bridge between theoretical models like the SYK and gravitational dual descriptions, deepening the understanding of holography, chaos, and hydrodynamic reformulations in lower-dimensional quantum systems.