Coadjoint orbit action of Virasoro group and two-dimensional quantum gravity dual to SYK/tensor models

TL;DR

The paper proposes a holographic bulk dual for the low-energy, Nambu-Goldstone sector of SYK-type models based on the coadjoint orbit action of the Virasoro group, realized as a covariant 2D gravity theory with a cosmological constant that admits asymptotically AdS2 geometries. The dynamics of the boundary gravitons, obtained from large diffeomorphisms, reduce to a Schwarzian action, reproducing the characteristic SYK soft-mode sector and its thermodynamics in a suitable double-scaling limit. By coupling bulk matter perturbatively, the authors outline a route to incorporate higher SYK modes and discuss how the bulk free energy, entropy, and specific heat qualitatively align with SYK predictions, while highlighting open questions about UV completion and the full spectrum. The work provides a concrete geometric framework connecting coadjoint Diff dynamics to 2D gravity and Schwarzian quantum mechanics, with implications for chaos, information scrambling, and the holographic description of tensor models.Overall, the study supports a Virasoro-coadjoint-orbit-based bulk dual for the IR sector of SYK-type theories and charts a path toward incorporating more of the SYK spectrum in a two-dimensional gravitational setting.

Abstract

The Nambu-Goldstone (NG) bosons of the SYK model are described by a coset space Diff/$\mathbb{SL}(2,\mathbb{R})$, where Diff, or Virasoro group, is the group of diffeomorphisms of the time coordinate valued on the real line or a circle. It is known that the coadjoint orbit action of Diff naturally turns out to be the two-dimensional quantum gravity action of Polyakov without cosmological constant, in a certain gauge, in an asymptotically flat spacetime. Motivated by this observation, we explore Polyakov action with cosmological constant and boundary terms, and study the possibility of such a two-dimensional quantum gravity model being the AdS dual to the low energy (NG) sector of the SYK model. We find strong evidences for this duality: (a) the bulk action admits an exact family of asymptotically AdS$_2$ spacetimes, parameterized by Diff/$\mathbb{SL}(2,\mathbb{R})$, in addition to a fixed conformal factor of a simple functional form; (b) the bulk path integral reduces to a path integral over Diff/$\mathbb{SL}(2,\mathbb{R})$ with a Schwarzian action; (c) the low temperature free energy qualitatively agrees with that of the SYK model. We show, up to quadratic order, how to couple an infinite series of bulk scalars to the Polyakov model and show that it reproduces the coupling of the higher modes of the SYK model with the NG bosons.

Figures (3)

• Figure 1: In the left panel, the top curve represents the Diff($R$)-orbit (or a Diff($S^1$)-orbit at finite temperature), at the IR fixed point$J= \infty$, of the classical large $N$ solution for the fermion bilocal $G_0(\tau_1, \tau_2) \sim (\tau_1- \tau_2)^{-2 \Delta}$; this represents the Nambu-Goldstones of Diff($R$)/$\mathbb{SL}(2,\mathbb{R})$ . The lower curve represents the orbit of a deformed solution $G_0'$ slightly away from the IR fixed point, with a small positive $1/J$. In the right panel, the top curve represents the orbit of the AdS$_2$ spacetime (these are asymptotically AdS$_2$ spacetimes, the two-dimensional equivalent of Brown-Henneaux geometries, which we will describe explicitly in Section \ref{['sec:asymp-ads']}). The bottom curve represents the orbit of a slightly deformed AdS$_2$ spacetime NAdS$_2$, with a controlled non-normalizable deformation (see \ref{['sec:asymp-ads']}).
• Figure 2: The difference in bulk action between AdS$_{\text{2}}$ and AAdS$_{\text{2}}$ geometries gets contribution only from the shaded region
• Figure :