Abstract
The Nambu-Goldstone (NG) bosons of the SYK model are described by a coset space Diff/SL(2, 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 AdS2 spacetimes, parameterized by Diff/SL(2, 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/SL(2, 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.
Highlights
Introduction and summaryThe Sachdev-Ye-Kitaev (SYK) model and other tensor models that have universal IR properties [1,2,3,4,5,6,7,8], are quantum mechanical models of large N fermionic particles, described by a Hamiltonian which, for Euclidean time τ = it, can be viewed alternatively as a onedimensional statistical model of fermions
The Nambu-Goldstone (NG) bosons of the SYK model are described by a coset space Diff /SL(2, R), where Diff, or Virasoro group, is the group of diffeomorphisms of the time coordinate valued on the real line or a circle
We find strong evidences for this duality: (a) the bulk action admits an exact family of asymptotically AdS2 spacetimes, parameterized by Diff /SL(2, 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 /SL(2, R) with a Schwarzian action; (c) the low temperature free energy qualitatively agrees with that of the SYK model
Summary
As explained in [1, 5], and briefly mentioned in the Introduction, the zero modes of the SYK model at the IR fixed point (we suggestively call these the Nambu-Goldstone (NG) modes, they differ somewhat from their higher dimensional counterpart, as explained below) are given by Diff transforms of the large N condensate of the bilocal ‘meson’ variable G(τ1, τ2) = ψI (τ1)ψI (τ2),. In terms of the bulk dual described by (2.7), that the central charge of the two-dimensional realization is proportional to N .5 In higher dimensions, such as in pion physics, the elements of the coset represent Nambu-Goldstone bosons, with kinetic terms given by a nonlinear sigma model (see, e.g. the discussion of pions in [39], chapter 19). In spite of the appearance of the derivatives, the above is a ‘potential’ term for the zero modes, similar to a pion mass term.
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