Solving the Schwarzian via the conformal bootstrap

By Tom Mertens, G. Turiaci, H. Verlinde. 2017

348 citations recorded by the source corpus.

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

Subfields: Atomic and Molecular Physics, and Optics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/50d9a9d2ade915997ac0e2dff8c9a81785c5813f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

1. D. Stanford
2. Phil Saad
3. Zhenbin Yang
4. S. Shenker
5. E. Witten
6. K. Jensen
7. Jordan S. Cotler
8. G. Turiaci

In Nuclear and High Energy Physics

1. D. Stanford
2. Phil Saad
3. Zhenbin Yang
4. Jordan S. Cotler
5. G. Turiaci
6. K. Jensen
7. S. Shenker
8. E. Witten

In Statistical and Nonlinear Physics

1. D. Stanford
2. Phil Saad
3. Jordan S. Cotler
4. K. Jensen
5. G. Turiaci
6. Zhenbin Yang
7. E. Witten
8. S. Shenker

Citations in the corpus

• The quantum gravity dynamics of near extremal black holes (2018, methodology): “y is controlled by its extrinsic curvature. Our goal will be to quantize this action and to provide expression for the full quantum gravity correlators of (1.1). This problem was considered before in [4, 5, 6, 7, 8] from various points of view. Here we will add one other point of view where we reduc”
• The statistical mechanics of near-BPS black holes (2020, methodology): “d in this paper, the Schwarzian path integral, we will only need the massless representations in the NS sector, due to the fact that we want the Schwarzian fermions to be antiperiodic as explained in [68].21 General representations are labeled by h, the eigenvalue of L 0, and ‘, the spin of the SU(2”
• A theory of reparameterizations for AdS3 gravity (2018, methodology): “(Other work relating Liouville theory to the Schwarzian model can be found in [15, 32, 33].”
• JT gravity and the ensembles of random matrix theory (2019, methodology): “The contribution of a particular topology to the path (1)This has included many different derivations of the exact path integral [12, 13, 14, 15, 16, 17, 18, 19, 20], studies of the Hilbert space structure [21, 18, 22, 23], correlation functions [24, 25, 26], the physical symmetry charges [27], and ”
• Statistical mechanics of a two-dimensional black hole (2018, methodology): “Two different reductions of the Schwarzian theory to a 2D CFT with a large central charge were proposed in [13].”
• The Schwarzian theory — origins (2018, methodology): “And indeed, the SL(2,R) particle-on-a-group action is equivalent to the Schwarzian action [27].”
• Late Time Correlation Functions, Baby Universes, and ETH in JT Gravity (2019, background): “In JT gravity, these effects are captured by the coupling of the matter to the boundary [20, 24, 25, 23, 43].”
• JT gravity as a matrix integral (2019, methodology): “The problem can be fixed [23, 91, 92, 90] by following the steps [93] used to impose the asymptotic boundary conditions for AdS3 [94] in the Chern-Simons formulation [95, 96] (see also [97, 98] for recent 3d discussion).”
• Low-dimensional de Sitter quantum gravity (2019, methodology): “Using a relation between the Schwarzian theory and two-dimensional Liouville theory the authors of [46] obtained integral expressions for gravitational scattering in nearly AdS2 which hold to all orders in perturbation theory in G2.”
• Towards a full solution of the large N double-scaled SYK model (2018, methodology): “We will find it particularly useful to compare our results to [28].”