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The DSSYK Model: Charge and Holography

Mattia Arundine

TL;DR

The work investigates how holography can extend beyond AdS by connecting the IR sector of SYK-type models to near-horizon gravity and de Sitter-like spacetimes. It develops the DSSYK framework and its charged variant, derives the Schwarzian IR action from reparametrization modes, and constructs a sine-dilaton bulk dual guiding a de Sitter–like holographic dictionary. Key results include explicit chord-diagram techniques for DSSYK, a detailed SD-equation analysis in the double-scaled limit, and a proposed bulk dual with boundary conditions that encode the DSSYK data, illuminating potential holographic descriptions of cosmological spacetimes. The findings advance the understanding of holography in non-AdS backgrounds and provide concrete calculational tools for correlators, spectral data, and phase structure in both the boundary and bulk theories.

Abstract

The Anti-de Sitter/Conformal Field Theory correspondence (AdS/CFT) is one of the most significant findings in theoretical physics and forms the basis of this thesis. Although highly powerful, the main limitation of AdS/CFT is that AdS does not appear in the real world outside of very specific limits. This limitation justifies the attempt to generalize the holographic principle to other spacetimes. In this thesis, we will pursue this direction and seek spacetimes that have at least some connection to de Sitter (dS), whose cosmological interest is evident. dS is, in fact, not only the geometry that best represents our Universe on large scales in the present, but also during the inflationary epoch that followed the Big Bang, where our current description of Nature fails completely. First, we will present some basic facts about the AdS/CFT correspondence. Then, the gravitational path integral will be introduced. After presenting the Sachdev-Ye-Kitaev (SYK) model and dilaton-gravity models, a holographic link between the two will be established. Next, we will discuss the double-scaled limit of SYK, known as DSSYK. We will then consider a ``charged'' variation of SYK with Dirac fermions, for which we will determine the fermionic two-point function. After studying the thermodynamic properties of the gravitational dual of DSSYK and the quasinormal modes of massive real scalars propagating in this geometry, we will conjecture how to modify the duality when considering the Dirac version of the model, showing that several bounds constrain the space of possible dual theories. Finally, we will summarize our findings and present an outlook on possible future developments based on the results described here.

The DSSYK Model: Charge and Holography

TL;DR

The work investigates how holography can extend beyond AdS by connecting the IR sector of SYK-type models to near-horizon gravity and de Sitter-like spacetimes. It develops the DSSYK framework and its charged variant, derives the Schwarzian IR action from reparametrization modes, and constructs a sine-dilaton bulk dual guiding a de Sitter–like holographic dictionary. Key results include explicit chord-diagram techniques for DSSYK, a detailed SD-equation analysis in the double-scaled limit, and a proposed bulk dual with boundary conditions that encode the DSSYK data, illuminating potential holographic descriptions of cosmological spacetimes. The findings advance the understanding of holography in non-AdS backgrounds and provide concrete calculational tools for correlators, spectral data, and phase structure in both the boundary and bulk theories.

Abstract

The Anti-de Sitter/Conformal Field Theory correspondence (AdS/CFT) is one of the most significant findings in theoretical physics and forms the basis of this thesis. Although highly powerful, the main limitation of AdS/CFT is that AdS does not appear in the real world outside of very specific limits. This limitation justifies the attempt to generalize the holographic principle to other spacetimes. In this thesis, we will pursue this direction and seek spacetimes that have at least some connection to de Sitter (dS), whose cosmological interest is evident. dS is, in fact, not only the geometry that best represents our Universe on large scales in the present, but also during the inflationary epoch that followed the Big Bang, where our current description of Nature fails completely. First, we will present some basic facts about the AdS/CFT correspondence. Then, the gravitational path integral will be introduced. After presenting the Sachdev-Ye-Kitaev (SYK) model and dilaton-gravity models, a holographic link between the two will be established. Next, we will discuss the double-scaled limit of SYK, known as DSSYK. We will then consider a ``charged'' variation of SYK with Dirac fermions, for which we will determine the fermionic two-point function. After studying the thermodynamic properties of the gravitational dual of DSSYK and the quasinormal modes of massive real scalars propagating in this geometry, we will conjecture how to modify the duality when considering the Dirac version of the model, showing that several bounds constrain the space of possible dual theories. Finally, we will summarize our findings and present an outlook on possible future developments based on the results described here.
Paper Structure (56 sections, 653 equations, 35 figures)

This paper contains 56 sections, 653 equations, 35 figures.

Table of Contents

  1. Introduction
  2. A (Very Brief) Introduction to AdS/CFT
  3. The GKPW Dictionary
  4. Holographic Renormalization Group
  5. Gravitational Path Integral
  6. Spacetime Thermodynamics
  7. First Example: the Schwarzschild Black Hole
  8. Second Example: Thermal de Sitter
  9. Third Example: Black Holes in AdS5
  10. The SYK Model
  11. Large N and Perturbation Theory
  12. The (G,S) Formalism
  13. Large p Expansion
  14. The Conformal Limit
  15. IR Effective Action
  16. ...and 41 more sections

Figures (35)

  • Figure 1: The Euclidean "cigar", with the $(d-2)$-sphere suppressed. The manifold caps off at $r_0$, so there is no inner (outer) horizon region and the manifold is regular there. The domain of the radial coordinate is $r \geq r_0$ for event horizons and $r \leq r_0$ for cosmological horizons.
  • Figure 2: Dependence of the radii of the "big" and "small" black holes in AdS$_5$ on the inverse temperature $\beta$, using dimensionless units. Big black holes always have a bigger radius than $r_*$, whereas small black holes always have a smaller one.
  • Figure 3: Any state can be defined through its overlap with a basis. A thermal state is characterized by its matrix elements $\langle \phi_1 | \rho | \phi_2 \rangle$ and can therefore be visualized as a "cylinder" at whose extremities one inserts any "in" and "out" state, which are connected through an Euclidean time evolution (that is, a path integral).
  • Figure 4: Left: Maximally extended Penrose diagram of an extremal black hole. The infinite chain signals the succession of black hole and white hole horizons that connect different universes. The vertical zigzagging line on the left is the singularity, spacelike infinities are on the right. The blue region is the AdS$_2$ near-horizon region, while the red dashed one is the patch covered by the Poincaré coordinates. Right: Penrose diagram and coordinates of global AdS$_2$, which possesses two boundaries. The Poincaré patch is the light yellow region, while the dark yellow one is the Rindler patch.
  • Figure 5: Left: Coordinates on the hyperbolic disk. $z$ is not shown, but it is $0$ on the boundary of the disk and increases towards $+\infty$ when going inwards. Right: A cutout from the hyperbolic disk.
  • ...and 30 more figures