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Correlators in the $\mathcal{N}=2$ Supersymmetric SYK Model

Cheng Peng, Marcus Spradlin, Anastasia Volovich

TL;DR

The paper analyzes correlation functions in the one-dimensional ${\cal N}=2$ supersymmetric SYK model, demonstrating that 4-point functions require a complete basis of both symmetric and antisymmetric conformal eigenfunctions. Using ladder diagram resummations in the conformal basis, the authors derive the diagonal and non-diagonal 4-point kernels, and show that the operator spectrum organizes into two ${\cal N}=2$ supermultiplets, with dimensions forming triplets around a common $h$. The retarded kernel analysis reveals maximal chaos, with the Lyapunov exponent saturating the bound via a pole at $h=-1$, and divergences in the 4-point functions arising from exchange of the full ${\cal N}=2$ multiplet, consistent with a super-Schwarzian effective action. The work also outlines a path to extract OPE data from 6-point functions and highlights the supersymmetric structure that constrains correlations across fermionic and bosonic sectors. Overall, the study extends the SYK toolbox to ${\cal N}=2$ supersymmetry, providing a detailed spectral and dynamical picture in terms of conformal eigenfunctions and ladder sums.

Abstract

We study correlation functions in the one-dimensional $\mathcal{N}=2$ supersymmetric SYK model. The leading order 4-point correlation functions are computed by summing over ladder diagrams expanded in a suitable basis of conformal eigenfunctions. A novelty of the $\mathcal{N}=2$ model is that both symmetric and antisymmetric eigenfunctions are required. Although we use a component formalism, we verify that the operator spectrum and 4-point functions are consistent with $\mathcal{N}=2$ supersymmetry. We also confirm the maximally chaotic behavior of this model and comment briefly on its 6-point functions.

Correlators in the $\mathcal{N}=2$ Supersymmetric SYK Model

TL;DR

The paper analyzes correlation functions in the one-dimensional supersymmetric SYK model, demonstrating that 4-point functions require a complete basis of both symmetric and antisymmetric conformal eigenfunctions. Using ladder diagram resummations in the conformal basis, the authors derive the diagonal and non-diagonal 4-point kernels, and show that the operator spectrum organizes into two supermultiplets, with dimensions forming triplets around a common . The retarded kernel analysis reveals maximal chaos, with the Lyapunov exponent saturating the bound via a pole at , and divergences in the 4-point functions arising from exchange of the full multiplet, consistent with a super-Schwarzian effective action. The work also outlines a path to extract OPE data from 6-point functions and highlights the supersymmetric structure that constrains correlations across fermionic and bosonic sectors. Overall, the study extends the SYK toolbox to supersymmetry, providing a detailed spectral and dynamical picture in terms of conformal eigenfunctions and ladder sums.

Abstract

We study correlation functions in the one-dimensional supersymmetric SYK model. The leading order 4-point correlation functions are computed by summing over ladder diagrams expanded in a suitable basis of conformal eigenfunctions. A novelty of the model is that both symmetric and antisymmetric eigenfunctions are required. Although we use a component formalism, we verify that the operator spectrum and 4-point functions are consistent with supersymmetry. We also confirm the maximally chaotic behavior of this model and comment briefly on its 6-point functions.

Paper Structure

This paper contains 14 sections, 67 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Operator Spectrum
  3. 2-point functions
  4. The diagonal 4-point kernel
  5. The non-diagonal 4-point kernels
  6. The chaotic behavior
  7. Symmetric and Antisymmetric Conformal Eigenfunctions
  8. Evaluating the 4-point Functions
  9. Setup
  10. The $\psi \bar{\psi} \psi \bar{\psi}$ and $b \bar{b} \psi \bar{\psi}$ 4-point functions
  11. The $\psi\bar{\psi}b\bar{b}$ and $b \bar{b} b \bar{b}$ 4-point functions
  12. The $\psi b \bar{\psi} \bar{b}$ 4-point function
  13. 6-point Functions
  14. Useful Integrals

Figures (3)

  • Figure 1: Kernel of the $\langle\psi_i(\tau_1)b_i(\tau_2)\bar{\psi}_j(\tau_3)\bar{b}_j(\tau_4)\rangle$ correlation. Iterating this kernel generates all ladder diagrams, which dominate the large-$N$ limit of the connected 4-point function.
  • Figure 2: Kernels of the $\langle\psi_i(\tau_1)\bar{\psi}_i(\tau_2){\psi}_j(\tau_3)\bar{\psi}_j(\tau_4)\rangle$, $\langle\psi_i(\tau_1)\bar{\psi}_i(\tau_2) {b}_j(\tau_3)\bar{b}_j(\tau_4)\rangle$, $\langle b_i(\tau_1)\bar{b}_i(\tau_2) {\psi}_j(\tau_3)\bar{\psi}_j(\tau_4)\rangle$ and $\langle b_i(\tau_1)\bar{b}_i(\tau_2) {b}_j(\tau_3)\bar{b}_j(\tau_4)\rangle$ correlation functions. Iterating these kernels to build ladder diagrams amounts to $2 \times 2$ matrix multiplication.
  • Figure 3: A plot of the eigenvalues for the $q=3$ model. Each figure illustrates one tower of ${\cal N}=2$ supermultiplets. Their intersects with the horizontal $k=1$ line give the dimensions of operators running in the OPE channel.