Operator approach to boundary Liouville theory

2018

151

218

In this paper we further study the 1d Schwarzian theory, the universal lowenergy limit of Sachdev-Ye-Kitaev models, using the link with 2d Liouville theory. We provide a path-integral derivation of the structural link between both theories, and study the relation between 3d gravity, 2d Jackiw-Teitelboim gravity, 2d Liouville and the 1d Schwarzian. We then generalize the Schwarzian double-scaling limit to rational models, relevant for SYK-type models with internal symmetries. We identify the holographic gauge theory as a 2d BF theory and compute correlators of the holographically dual 1d particleon-a-group action, decomposing these into diagrammatic building blocks, in a manner very similar to the Schwarzian theory.

Section: Path Integral Derivation Of Schwarzian Correlatorsmentioning

confidence: 99%

“…Next we define this theory on a cylindrical surface between two ZZ-branes [45] at σ = 0 and σ = π ( Figure 2 The classical solution of this configuration is well-known [38,39]:…”

Section: Gervais-neveu Field Transformationmentioning

confidence: 99%

The Schwarzian theory — origins

2018

151

218

“…The classical solution of Liouville field with the above ZZ-brane boundary conditions is expressed in terms of the single function f via [37,38]…”

Section: Schwarzian Qm As a Limit Of Virasoro Cftmentioning

confidence: 99%

Shockwave S-matrix from Schwarzian quantum mechanics

Lam

Turiaci

et al. 2018

118

158

Schwarzian quantum mechanics describes the collective IR mode of the SYK model and captures key features of 2D black hole dynamics. Exact results for its correlation functions were obtained in JHEP 1708, 136 (2017) [arXiv:1705.08408]. We compare these results with bulk gravity expectations. We find that the semi-classical limit of the OTO four-point function exactly matches with the scattering amplitude obtained from the Dray-'t Hooft shockwave S-matrix. We show that the two point function of heavy operators reduces to the semi-classical saddle-point of the Schwarzian action. We also explain a previously noted match between the OTO four point functions and 2D conformal blocks. Generalizations to higher-point functions are discussed.

“…The boundary conditions of φ are that the regions near σ = 0 and σ = π corresponds to the asymptotic regions of a hyperbolic cylinder. It is shown in [50,51] that the lowest energy solution is 4µ 4 e 2φ = sin −2 σ. Written in the form ds 2 = e 2φ dudv this describes a hyperbolic geometry of the form shown in figure 4.…”

Section: Zz Branes and Kinematic Spacementioning

confidence: 99%

Solving the Schwarzian via the conformal bootstrap

Turiaci

Verlinde

2017

317

632

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 N = 1 and N = 2 supersymmetric Schwarzian QM.