# Solving the Schwarzian via the Conformal Bootstrap

## TL;DR

This paper derives exact correlation functions for the Schwarzian theory, revealing connections to 2D Virasoro CFT, and explores implications for black hole physics and supersymmetric extensions.

## Contribution

It provides a novel integral representation for Schwarzian correlators using a 2D CFT framework and introduces a crossing kernel (R-matrix) related to gravitational scattering.

## Key findings

- Exact correlation functions expressed as momentum space integrals
- Identification of the R-matrix as a gravitational scattering amplitude
- Extension of results to supersymmetric Schwarzian models

## 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.