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A line of CFTs: from generalized free fields to SYK

David J. Gross, Vladimir Rosenhaus

TL;DR

The paper introduces conformal SYK (cSYK), a line of fixed points obtained by replacing the UV sector of SYK with a bilocal term that preserves $SL(2,\mathbb{R})$ invariance for all couplings. It provides a concrete bilocal action derived from a local tower of auxiliary fields and explains its AdS$_2$ bulk interpretation as a non-gravitational theory on a fixed background. Across the line, the authors compute two- and three-point functions of $O(N)$ bilinears, showing interpolation between generalized free-field behavior and SYK infrared physics, with special attention to the operator $\mathcal{O}_0$ and its role in conformal symmetry. The discussion contrasts the bulk duals of cSYK and SYK, highlighting the absence of a dilaton in cSYK and the JT gravity structure in SYK, and explains how the line of fixed points accounts for observed relations in bilinear correlators at large $q$.

Abstract

We point out that there is a simple variant of the SYK model, which we call cSYK, that is $SL(2,R)$ invariant for all values of the coupling. The modification consists of replacing the UV part of the SYK action with a quadratic bilocal term. The corresponding bulk dual is a non-gravitational theory in a rigid AdS$_2$ background. At weak coupling cSYK is a generalized free field theory; at strong coupling, it approaches the infrared of SYK. The existence of this line of fixed points explains the previously found connection between the three-point function of bilinears in these two theories at large $q$.

A line of CFTs: from generalized free fields to SYK

TL;DR

The paper introduces conformal SYK (cSYK), a line of fixed points obtained by replacing the UV sector of SYK with a bilocal term that preserves invariance for all couplings. It provides a concrete bilocal action derived from a local tower of auxiliary fields and explains its AdS bulk interpretation as a non-gravitational theory on a fixed background. Across the line, the authors compute two- and three-point functions of bilinears, showing interpolation between generalized free-field behavior and SYK infrared physics, with special attention to the operator and its role in conformal symmetry. The discussion contrasts the bulk duals of cSYK and SYK, highlighting the absence of a dilaton in cSYK and the JT gravity structure in SYK, and explains how the line of fixed points accounts for observed relations in bilinear correlators at large .

Abstract

We point out that there is a simple variant of the SYK model, which we call cSYK, that is invariant for all values of the coupling. The modification consists of replacing the UV part of the SYK action with a quadratic bilocal term. The corresponding bulk dual is a non-gravitational theory in a rigid AdS background. At weak coupling cSYK is a generalized free field theory; at strong coupling, it approaches the infrared of SYK. The existence of this line of fixed points explains the previously found connection between the three-point function of bilinears in these two theories at large .

Paper Structure

This paper contains 9 sections, 83 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Bilocal Action
  3. A Tower of Auxiliary Fields
  4. A Line of Fixed Points
  5. Correlation Functions
  6. Discussion
  7. OPE and Three-Point Function of Bilinears
  8. OPE
  9. Three-point Function Bilinears

Figures (1)

  • Figure 1: (a) A plot of the running of the normalization of the cSYK fermion two-point function, ${\overline b}$, as a function of $J$, for $q=8,6,4$. The larger the $q$, the slower the decay. (b) A plot of ${\overline b}^q$; this decreases with increasing $q$ (the opposite of ${\overline b}$).