Holographic dual to charged SYK from 3D gravity and Chern-Simons

Abstract

In this paper, we obtain a bulk dual to SYK model, including SYK model with U (1) charge, by Kaluza-Klein (KK) reduction from three dimensions. We show that KK reduction of the 3D Einstein action plus its boundary term gives the Jackiw-Teitelboim (JT) model in 2D with the appropriate 1D boundary term. The size of the KK radius gets identified with the value of the dilaton in the resulting near-AdS2 geometry. In presence of U(1) charge, the 3D model additionally includes a U (1) Chern-Simons (CS) action. In order to describe a boundary theory with non-zero chemical potential, we also introduce a coupling between CS gauge field and bulk gravity. The 3D CS action plus the new coupling term with appropriate boundary terms reduce in two dimensions to a BF-type action plus a source term and boundary terms. The KK reduced 2D theory represents the soft sector of the charged SYK model. The pseudo-Nambu-Goldstone modes of combined Diff/SL(2, R) and U (1) local /U (1) transformations are represented by combined large diffeomorphisms and large gauge transformations. The effective action of the former is reproduced by the action cost of the latter in the bulk dual, after appropriate identification of parameters. We compute chaotic correlators from the bulk and reproduce the result that the contribution from the "boundary photons" corresponds to zero Liapunov exponent.

Notes on the complex Sachdev-Ye-Kitaev model

“…Our computation of S should be contrasted with that for higher-dimensional charged black holes [36,5,3,[13][14][15][16][17][18], summarized in Appendix G. In the latter case, the value of S in Eq. (G.4) is determined by the horizon area and has no direct connection to the parameters of the SYK model.…”

“…We denote the T = 0 radius of the black hole by R h . Then we will discuss the universal structure of the theory of such black holes at temperature T 1/R h [3,[13][14][15][16][17][18], where the effective action in Eq. (1.12) applies.…”

“…An analogous effective theory of the complex SYK model also includes a U(1) mode [8] (1. 13) In this effective theory, the U(1) and SL(2, R) freedom are actually decoupled, which can be demonstrated by the variable change λ(τ ) = λ(τ ) + iE 2π β τ − ϕ(τ ) . The action (1.12) is characterized by two parameters, K and γ, and these can be specified by their connection to thermodynamics.…”

Gravitational anomalies in nAdS2/nCFT1

“…It is easier to first focus on the contributions from EMD, which resembles the analysis in [24] adopted to our setup. As in the UV analysis, the Gibbons-Hawking term guarantees a Dirichlet boundary problem 31) and the counterterm that renders the variation finite is…”

“…The resulting 2D theory of gravity contains, in addition to the metric, a gauge field and a dilaton as expected in Kaluza-Klein theory. Our work builds upon the developments in [17,30], where the 2D holographic dictionary was studied by dimensionally reducing the 3D Einstein-Hilbert action with a negative cosmological constant.Other relevant work includes [31], which focuses on the effects of U (1) Chern-Simons fields in 2D; see also [22,[32][33][34][35].One question we investigate is the relation between the nCFT 1 , that describes the near horizon physics of near-extremal BTZ, to the parent CFT 2 , that is dual to AdS 3 . This relation is subtle:The conformal (Weyl) anomaly in AdS 3 /CFT 2 can easily be confused with the anomaly appearing…”