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Towards a 2d QFT Analog of the SYK Model

Gustavo Turiaci, Herman Verlinde

TL;DR

This work constructs a 2D quantum field theory that extends the SYK paradigm by combining a UV topological Ising CFT with a random IR interaction, yielding an IR conformal phase with a nonlinearly realized reparametrization mode. The emergent Goldstone mode is governed by a double Schwarzian action, which can be recast as the boundary action of 3D AdS gravity, establishing a holographic link via a finite-cutoff AdS$_3$ setup and a TTbar-type deformation with coupling $\mu$ related to the central charge $c$. The analysis via Schwinger-Dyson equations shows a conformal IR fixed point with a zero-mode structure akin to the SYK model, and the four-point kernel reproduces the stress-tensor singularities, signaling maximal chaos in the 2D context. Together these results provide a concrete framework for holographic duality in two dimensions and offer a concrete bridge between IR conformal dynamics, TTbar deformations, and AdS$_3$ gravity, with implications for Liouville theory and potential higher-dimensional generalizations.

Abstract

We propose a 2D QFT generalization of the Sachdev-Ye-Kitaev model, which we argue preserves most of its features. The UV limit of the model is described by $N$ copies of a topological Ising CFT. The full interacting model exhibits conformal symmetry in the IR and an emergent pseudo-Goldstone mode that arises from broken reparametrization symmetry. We find that the effective action of the Goldstone mode matches with the 3D AdS gravity action, viewed as a functional of the boundary metric. We compute the spectral density and show that the leading deviation from conformal invariance looks like a $T \bar{T}$ deformation. We comment on the relation between the IR effective action and Liouville CFT.

Towards a 2d QFT Analog of the SYK Model

TL;DR

This work constructs a 2D quantum field theory that extends the SYK paradigm by combining a UV topological Ising CFT with a random IR interaction, yielding an IR conformal phase with a nonlinearly realized reparametrization mode. The emergent Goldstone mode is governed by a double Schwarzian action, which can be recast as the boundary action of 3D AdS gravity, establishing a holographic link via a finite-cutoff AdS setup and a TTbar-type deformation with coupling related to the central charge . The analysis via Schwinger-Dyson equations shows a conformal IR fixed point with a zero-mode structure akin to the SYK model, and the four-point kernel reproduces the stress-tensor singularities, signaling maximal chaos in the 2D context. Together these results provide a concrete framework for holographic duality in two dimensions and offer a concrete bridge between IR conformal dynamics, TTbar deformations, and AdS gravity, with implications for Liouville theory and potential higher-dimensional generalizations.

Abstract

We propose a 2D QFT generalization of the Sachdev-Ye-Kitaev model, which we argue preserves most of its features. The UV limit of the model is described by copies of a topological Ising CFT. The full interacting model exhibits conformal symmetry in the IR and an emergent pseudo-Goldstone mode that arises from broken reparametrization symmetry. We find that the effective action of the Goldstone mode matches with the 3D AdS gravity action, viewed as a functional of the boundary metric. We compute the spectral density and show that the leading deviation from conformal invariance looks like a deformation. We comment on the relation between the IR effective action and Liouville CFT.

Paper Structure

This paper contains 16 sections, 79 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The 2D model
  3. Lagrangian formulation
  4. UV limit: Topological Ising CFT
  5. Hamiltonian formulation
  6. Schwinger-Dyson equations
  7. SD equations at large $N$
  8. Conformal limit
  9. Two-point function
  10. Four-point function
  11. Effective action of the Goldstone mode
  12. Double Schwarzian action
  13. Free energy and spectral density
  14. Relation with AdS${}_3$ gravity
  15. Conclusion
  16. ...and 1 more sections

Figures (4)

  • Figure 1: In a topological CFT, local operators are attached to two Wilson lines that connect to past null infinity. Whether two operators are space-like or time-like separated is a topological distinction, encoded via the relative ordering of the asymptotic end-points $x^\pm_1$ and $x^\pm_2$ of the respective Wilson lines. The lines are shown with zigzags to indicate that the bulk has no fixed metric.
  • Figure 2: Diagrammatic representation of the SD equations \ref{['sdo']}, \ref{['sdt']} and \ref{['sdf']} for $q=2$.
  • Figure 3: Diagrammatic definition of the kernel that gives the four-fermion correlation function. Here each line represents multiple dressed propagators, with multiplicity as indicated.
  • Figure 4: Plot of $\det(\mathbb{1} - K)$ as a function of the left scale dimension $h$ with $\bar{h}=0$. The dashed magenta plot corresponds to $\Delta = 1/4$ and $s=1/2$, and the blue plot to $\Delta = s = 1/4$.