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

TL;DR

This work builds a holographic bulk dual for the low-energy, soft sector of the charged SYK model by performing a Kaluza-Klein reduction of 3D gravity with a negative cosmological constant and a U(1) Chern-Simons term. The resulting 2D theory combines Jackiw-Teitelboim gravity with a BF-type gauge sector; the dilaton arises as the KK radius, and the boundary dynamics comprise Schwarzian reparameterizations together with a U(1) soft mode, yielding a Schwarzian plus sigma-model effective action that matches the field theory description. A careful treatment of boundary terms and counterterms provides a finite effective action for pseudo-Nambu-Goldstone modes, and the bulk computation of OTO correlators shows that boundary photons do not contribute to chaos, while gravitons generate maximal chaos. The construction clarifies how the charged SYK soft sector can be captured by a bulk dual and establishes explicit parameter mappings between bulk and boundary theories. This framework sharpens our understanding of how U(1) charge modifies the SYK/JT correspondence and the nature of quantum chaos in holographic systems.

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-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.

Figures (2)

• Figure 1: The left panel represents the symmetry orbit of the classical large $N$ solution for the fermion bilocal $G_0(\tau_1, \tau_2)$, in the charged SYK model at the IR fixed point $J= \infty$. The symmetry group, $\cal G$ is defined as the group of transformations generated by Diff and local U(1) transformations. The classical solution is invariant under ${\cal H}$= $SL(2,R) \times U(1)_{global}$$\in$$\cal G$. The orbit represents the Nambu Goldstone modes belonging to the coset ${\cal G}/{\cal H}$. The right panel represents the symmetry orbit of AdS$_{\text{2}}$ black hole under large diffeomorphisms. Points of the orbit parameterize the same coset as on the left, and are represented by the asymptotically AdS$_2$ spacetimes with horizon (\ref{['aads2intro']}) and $A_\tau$ given by (\ref{['eq:generalAt']})
• Figure 2: The picture in the right panel (asymptotically AdS$_2$ metric with straight boundary) is achieved by performing large diffeomorphisms (\ref{['eq:exact-large-diff']}) on the left panel (AdS$_2$/AdS$_{\text{2}}$ black hole metric with wiggly boundary)