A universal Schwarzian sector in two-dimensional conformal field theories

TL;DR

This work demonstrates that a universal Schwarzian sector governs a wide class of 2D CFTs at large central charge and low temperature, even without a holographic dual. By combining conformal bootstrap in the grand-canonical ensemble with modular and fusion kernel techniques, the authors show that the density of states and all correlators reduce to Schwarzian dynamics in the near-extremal regime, with controlled corrections. They provide a detailed bridge between 2D CFT data and a gravitational interpretation in terms of a near-horizon AdS$_2$ throat described by JT gravity, including precise results for two- and higher-point functions and for OTOCs that saturate the chaos bound. The analysis applies to general irrational CFTs with a twist gap and shows the universality of Schwarzian physics beyond conventional AdS/CFT setups, illuminating how gravity-like chaotic dynamics emerge in broad quantum systems.

Abstract

We show that an extremely generic class of two-dimensional conformal field theories (CFTs) contains a sector described by the Schwarzian theory. This applies to theories with no additional symmetries and large central charge, but does not require a holographic dual. Specifically, we use bootstrap methods to show that in the grand canonical ensemble, at low temperature with a chemical potential sourcing large angular momentum, the density of states and correlation functions are determined by the Schwarzian theory, up to parametrically small corrections. In particular, we compute out-of-time-order correlators in a controlled approximation. For holographic theories, these results have a gravitational interpretation in terms of large, near-extremal rotating BTZ black holes, which have a near horizon throat with nearly AdS$_2 \times S^1$ geometry. The Schwarzian describes strongly coupled gravitational dynamics in the throat, which can be reduced to Jackiw-Teitelboim (JT) gravity interacting with a $U(1)$ field associated to transverse rotations, coupled to matter. We match the physics in the throat to observables at the AdS$_3$ boundary, reproducing the CFT results.

Figures (5)

• Figure 1: Diagram of the fusion transformation that was used to compute the left moving vacuum block in the appropriate limit. The blue lines correspond to the two operator insertions.
• Figure 2: Diagram of the fusion and modular transformation that was used to compute the left moving vacuum torus block in the appropriate limit. In the top diagrams, the circle is a spatial circle, and in the bottom diagrams it is a Euclidean time circle; these are swapped by the S-transform in step 2.
• Figure 3: Diagram of the fusion transformation that was used to compute the four-point left moving vacuum block in the appropriate limit. The blue (red) line corresponds to the external $\mathcal{O}_A$ ($\mathcal{O}_B$) insertion.
• Figure 4: Diagram of the braiding transformation that is need to compute out-of-time-ordered correlators.
• Figure 5: The near extremal BTZ geometry at fixed time, from the horizon (leftmost circle) to the asymptotically AdS$_3$ boundary (rightmost circle $\partial_3$). In region (A) the geometry is approximately nearly AdS$_2\times S^1$ (described by JT gravity), and in (B) the geometry is approximately extremal BTZ and the physics is classical. In the overlap between (A) and (B), we introduce the boundary $\partial_2$ (the blue line) where the Schwarzian mode lives.