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

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