# More on Supersymmetric and 2d Analogs of the SYK Model

## TL;DR

This paper develops supersymmetric and two-dimensional analogs of the SYK model, analyzing their conformal properties, four-point functions, and key physical quantities like chaos and anomalous dimensions, revealing new insights into higher-dimensional SYK-like theories.

## Contribution

It introduces new 2d supersymmetric and bosonic SYK analogs, clarifies conformal eigenfunctions, and computes physical quantities, advancing understanding of SYK models beyond 1d.

## Key findings

- Supersymmetric 2d SYK flows to a large N CFT with strong interactions.
- Derived integral formula for four-point function at order 1/N.
- Identified issues with 2d SYK-like CFTs possessing continuous global symmetries.

## Abstract

In this paper, we explore supersymmetric and 2d analogs of the SYK model. We begin by working out a basis of (super)conformal eigenfunctions appropriate for expanding a four-point function. We use this to clarify some details of the 1d supersymmetric SYK model. We then introduce new bosonic and supersymmetric analogs of SYK in two dimensions. These theories consist of $N$ fields interacting with random $q$-field interactions. Although models built entirely from bosons appear to be problematic, we find a supersymmetric model that flows to a large $N$ CFT with interaction strength of order one. We derive an integral formula for the four-point function at order $1/N$, and use it to compute the central charge, chaos exponent and some anomalous dimensions. We describe a problem that arises if one tries to find a 2d SYK-like CFT with a continuous global symmetry.