The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual

TL;DR

The paper analyzes the SYK model’s soft reparametrization mode and its gravity dual, deriving a non-local correction to the Schwarzian action arising from UV–IR interactions and showing its equivalence to a dilaton-gravity boundary action. It develops a replica-based formalism to treat disorder-averaged correlators, characterizes the conformal kernel spectrum, and computes leading and subleading four-point functions, including OTOCs with maximal chaos at low temperature. A detailed gravity-side construction using dilaton gravity with conformal matter reproduces the Schwarzian dynamics and the new non-local boundary term, providing a deeper link between SYK fluctuations and 2D gravity. The work raises open questions about the separation of soft and UV degrees of freedom, the interpretation of subleading corrections to chaos, and the precise bulk dual of the non-local corrections.

Abstract

We give an exposition of the SYK model with several new results. A non-local correction to the Schwarzian effective action is found. The same action is obtained by integrating out the bulk degrees of freedom in a certain variant of dilaton gravity. We also discuss general properties of out-of-time-order correlators.

Figures (6)

• Figure 1: ($1+1$)-dimensional black hole: a) Structure of space-time and infalling matter (shown as red dotted lines); b) The gravitational perturbations has turned into a "shock wave"; c) The same geometry in different coordinates.
• Figure 2: Diagrammatic calculus for the expansion of action \ref{['rdact1']} around the saddle point $(\widetilde{\Sigma}_*,\widetilde{G}_*)$ for $q=4$. Degenerate $2$-gon sheets representing $\widetilde{G}_*$ may be attached directly to the template tabs. When the building blocks are composed into a diagram, each orientation conflict between a sheet and an adjacent seam or between a sheet and the template gives a factor of $-1$.
• Figure 3: Cartoon of the window function $u$ dressing UV perturbations (left) and the response $\delta g=g_{\text{UV},I}$ to a UV perturbation $s_{I}$ (right) as a function of $\xi=\ln(|\theta_{12}|/\varepsilon)$.
• Figure 4: Map of the physical space (disk $D$) to the region $D'$ of the Poincare disk.
• Figure 5: To evaluate the non-local part of the action, we identify the domain of the Poincare disk with the upper half-plane and approximate $D'$ as shown on the right-hand-side.