Solving the Schwarzian via the Conformal Bootstrap

TL;DR

This work provides an exact, nonperturbative solution for finite-temperature correlation functions in the 1D Schwarzian quantum mechanics, seen as the low-energy limit of the SYK model. By embedding the Schwarzian in a large-c limit of 2D Virasoro CFT and employing ZZ brane techniques in Liouville theory, the authors derive explicit momentum-space amplitudes for two- and four-point functions and, crucially, the out-of-time-ordered correlator via a crossing (R-matrix) kernel. The R-matrix is shown to be a Schwarzian SU(1,1) 6j-symbol that encodes gravitational scattering near an AdS2 horizon and yields maximal Lyapunov growth in the chaotic regime. The paper also extends the framework to N=1 and N=2 supersymmetric Schwarzian QM, revealing universal spectral data, modular bootstrap structures, and the role of BPS versus non-BPS sectors. Overall, this work establishes a concrete 2D CFT/Liouville-based bootstrap route to exact Schwarzian correlators, illuminating holography for nearly AdS2 and chaotic dynamics in SYK-like systems.

Abstract

We obtain exact expressions for a general class of correlation functions in the 1D quantum mechanical model described by the Schwarzian action, that arises as the low energy limit of the SYK model. The answer takes the form of an integral of a momentum space amplitude obtained via a simple set of diagrammatic rules. The derivation relies on the precise equivalence between the 1D Schwarzian theory and a suitable large $c$ limit of 2D Virasoro CFT. The mapping from the 1D to the 2D theory is similar to the construction of kinematic space. We also compute the out-of-time ordered four point function. The momentum space amplitude in this case contains an extra factor in the form of a crossing kernel, or R-matrix, given by a 6j-symbol of SU(1,1). We argue that the R-matrix describes the gravitational scattering amplitude near the horizon of an AdS${}_2$ black hole. Finally, we discuss the generalization of some of our results to ${\cal N}=1$ and ${\cal N}=2$ supersymmetric Schwarzian QM.

Figures (12)

• Figure 1: Overview of different models with underlying $SL(2,\mathbb{R})$ symmetry. Red lines indicate one-way lines: they are projections that reduce the dimension of the phase space.
• Figure 2: The spectrum of states in the Schwarzian theory arises from the CFT spectrum of states with conformal dimension $\Delta = \frac{c-1}{24} + b^2 E$, in the limit $b\to 0$. The operators in the Schwarzian are all light CFT operators with conformal dimension $\Delta = \ell$.
• Figure 3: The identity character can be represented as the annulus partition sum of the Virasoro CFT, or by using channel duality, as the transition amplitude between two ZZ boundary states.
• Figure 4: Geometry of the classical Liouville background between two ZZ branes.
• Figure 5: The kinematic space of the Schwarzian theory. The bi-local operator (\ref{['bilocalo']}) in the 1D QM is represented by a local Liouville CFT vertex operator in the 2D bulk. The boundary of the kinematic space corresponds to the limit where the two end-points of the bi-local operator coincide.