Solving the Schwarzian via the conformal bootstrap

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

doi:10.1007/jhep08(2017)136

Cited by 357Black Holes and Theoretical PhysicsAlgebraic structures and combinatorial modelsNoncommutative and Quantum Gravity Theories

Solving the Schwarzian via the conformal bootstrap

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Mertens, Thomas G., Turiaci, Gustavo J., Verlinde, Herman L.·2017·Journal of High Energy Physics

DOI 10.1007/jhep08(2017)136

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Abstract

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 $$ \mathcal{N}=1 $$ and $$ \mathcal{N}=2 $$ supersymmetric Schwarzian QM.

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Partial results for boundary matter correlators dual to JT gravity coupled to matter were obtained in Bagrets et al. (2016, 2017), and the full solution for all correlators was derived in Mertens et al. (2017) by realizing that the Schwarzian theory is a limit of Liouville field theory.
“Some partial results were obtained in Bagrets et al. (2016, 2017) and the full solution for all correlators was derived in Mertens et al. (2017) realizing that the Schwarzian theory is a limit of Liouvile field theory, further elaborated in Mertens (2018); Lam et al. (2018).”
From Solvable models of quantum black holes: a review on Jackiw–Teitelboim gravity (2023) · cites this paper as Mertens et al. (2017)
The four-point function including shockwave interaction is given by inserting amplitude (3.36) in (3.34).
“The final result obtained in Mertens et al. (2017) is given by inserting in (3.34) the following amplitude:3.36”
From Solvable models of quantum black holes: a review on Jackiw–Teitelboim gravity (2023) · cites this paper as Mertens et al. (2017)
More complicated correlators can be computed by diagrammatic rules, which can compute any bilocal Schwarzian correlator or boundary correlators with quantum gravity effects.
“More complicated correlators can be computed by following a simple set of diagrammatic rules, developed in Mertens et al. (2017) and Lam et al. (2018). These can be used to compute any bilocal Schwarzian correlator, or equivalently the boundary correlators with quantum gravity effects.”
From Solvable models of quantum black holes: a review on Jackiw–Teitelboim gravity (2023) · cites this paper as Mertens et al. (2017)
Supersymmetric JT gravity has a known disk density of states with a universal small E behavior different from bosonic JT gravity's square root spectral edge.
“As an example consider supersymmetric JT gravity. Its disk density of states is known (Stanford and Witten 2017; Mertens et al. 2017)”
From Solvable models of quantum black holes: a review on Jackiw–Teitelboim gravity (2023) · cites this paper as Mertens et al. 2017

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Authors: Mertens, T. G., Turiaci, G. J., & Verlinde, H. L.
Affiliations: Princeton University (US)
Corresponding author: Mertens, Thomas G.
ORCID: Mertens, Thomas G.
Turiaci, Gustavo J.
Verlinde, Herman L.
Venue: Journal of High Energy Physics
Publisher: Springer Science and Business Media LLC
Published: 2017-08
Volume: 2017
Issue: 8
ISSN: 1029-8479 (ISSN-L)
DOI: 10.1007/jhep08(2017)136
OpenAlex: W2618869271
Cited by: 357
References: 58
Citations by year: 2026: 25 · 2025: 38 · 2024: 25 · 2023: 45 · 2022: 51 · 2021: 40 · 2020: 40 · 2019: 52 · 2018: 32 · 2017: 9
FWCI: 18.61
Type: journal article
Language: en
Open access: diamond
doi.org
Licence: cc-by
Locations: Journal of High Energy Physics (publishedVersion, cc-by)
arXiv (Cornell University) (publishedVersion)
Ghent University Academic Bibliography (Ghent University) (submittedVersion, cc-by)
DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
arXiv (Cornell University)
Funders: National Science Foundation (1620059)
Fonds Wetenschappelijk Onderzoek
Vlaamse regering
Topics: Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
Physical Sciences › Mathematics › Geometry and Topology › Algebraic structures and combinatorial models
Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
Keywords: Schwarzian action, conformal bootstrap, SYK model, one-dimensional quantum mechanics, large central charge, kinematic space, R-matrix, 6j symbols, SU(1,1), AdS2 black hole, gravitational scattering, out-of-time-order correlators, four-point functions

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Solving the Schwarzian via the conformal bootstrap

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