Solving the Schwarzian via the conformal bootstrap

Thomas G. Mertens, Gustavo J. Turiaci, Herman L. Verlinde

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214 Scopus citations


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 AdS2 black hole. Finally, we discuss the generalization of some of our results to N= 1 and N= 2 supersymmetric Schwarzian QM.

Original languageEnglish (US)
Article number136
JournalJournal of High Energy Physics
Issue number8
StatePublished - Aug 1 2017

All Science Journal Classification (ASJC) codes

  • Nuclear and High Energy Physics


  • 2D Gravity
  • AdS-CFT Correspondence
  • Conformal Field Theory
  • Field Theories in Lower Dimensions


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