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

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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 in random matrix theory. In a large N collective fields description, the ramp was reproduced by Saad et al. (A semiclassical ramp in SYK and in gravity, arXiv:1806.06840) by replica symmetry breaking saddles. We derive a two sides Schwarzians effective action for fluctuations around the ramp critical saddles, by computing responses to a smeared version of the two replica kinetic kernel. Our result confirms [1], where the form of the action was heuristically guessed by indirect arguments supported by numerical evidences.