# Derivation of the two Schwarzians effective action for the Sachdev-Ye-Kitaev spectral form factor

## TL;DR

This paper derives a two-sided Schwarzian effective action for fluctuations around the spectral form factor's ramp in the SYK model, confirming previous assumptions about locality and providing a deeper understanding of the model's non-linear responses.

## Contribution

It introduces a novel derivation of the two-sided Schwarzian effective action for the SYK spectral form factor, adapting methods from Kitaev and Suh to a two-replica system.

## Key findings

- Derived a two-sided Schwarzian effective action for SYK spectral form factor.
- Confirmed the form of the action previously assumed to be local.
- Enhanced understanding of non-linear responses in the SYK model.

## Abstract

The Sachdev-Ye-Kitaev model spectral form factor exhibits absence of information loss in the form of a ramp and a plateau, that are typical of random matrix theory. In a large $N$ collective fields description, the ramp was reproduced by Saad, Shenker and Stanford \cite{Saad:2018bqo}, by replica symmetry breaking saddles for a connected component of the analytically continued to real times thermal partition function two point function. We derive a two sides Schwarzians effective action for fluctuations around the ramp critical saddles, by adapting to the two replica system a method by Kitaev and Suh \cite{Kitaev:2017awl} for studying non linear responses to the conformal breaking kinetic operator in regular SYK. Our result confirms \cite{Saad:2018bqo}, where the form of the action was obtained by assuming locality.