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

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-AdS$_2$ 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 $\textit{Diff} /\mathbb{SL}(2,\mathbb{R})$ and $U(1)_{\rm local}/U(1)$ 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.