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

## TL;DR

This paper investigates the connection between two-dimensional quantum gravity described by the Polyakov action with a cosmological constant and the low-energy sector of the SYK model, proposing a duality with asymptotically AdS$_2$ spacetimes.

## Contribution

It demonstrates that the Polyakov action with boundary terms can serve as an AdS dual to the SYK model's NG bosons, including an exact family of asymptotically AdS$_2$ solutions and a matching low-temperature free energy.

## Key findings

- Bulk action admits asymptotically AdS$_2$ solutions parameterized by Diff/$ ext{SL}(2, ext{R})$
- Bulk path integral reduces to a Schwarzian action over Diff/$ ext{SL}(2, ext{R})$
- Low temperature free energy matches that of the SYK model

## 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.