Abstract
In this paper, we obtain a bulk dual to the low energy sector of the 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.
Highlights
Introduction and summaryFinding simple field theories dual to gravitational systems has been a long standing goal
The c = 1 matrix model was found to be governed by W∞ symmetries; the dynamics could be abstracted in terms of quantizing coadjoint orbits of W∞
The dilaton comes from the radius of the KK direction; as emphasized above, a small non-zero KK radius corresponds to a small symmetry breaking parameter of AdS2 symmetry, giving a geometric meaning to such a role played by the dilaton in the JT model
Summary
Finding simple field theories dual to gravitational systems has been a long standing goal. One has to introduce a small breaking of the reparameterization symmetry by being slightly away from the strong coupling fixed point; the dynamics of the pseudo NG modes is governed by an action described by the Schwarzian of the reparametrization function (see (1.2) below). The pseudo Nambu Goldstone modes of reparameterization invariance of the SYK model are reminiscent of large diffeomorphisms of an asymptotically AdS2 geometry. This idea has been implemented in the bulk dual, in the Jackiw-Teitelboim (JT) models [13, 14] as well as in the Polyakov gravity model [2], based on the coadjoint orbit point of view mentioned above. In our case, of KK reduction from 3D, it is rather different; the breaking of the AdS2 symmetry is measured by a non-zero KK radius
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