# Supersymmetric SYK Model: Bi-local Collective Superfield/Supermatrix Formulation

## TL;DR

This paper develops a bi-local superfield and supermatrix formulation for supersymmetric SYK models, simplifying their analysis and providing new tools for understanding their superconformal structure.

## Contribution

It introduces a bi-local superfield and supermatrix framework for SUSY SYK models, enabling systematic analysis and diagonalization of the quadratic action.

## Key findings

- Supermatrix formulation simplifies SUSY SYK analysis.
- Eigenvectors of superconformal Casimir identified.
- Quadratic action diagonalized in large N limit.

## Abstract

We discuss the bi-local collective theory for the $\mathcal{N}=1,2$ supersymmetric Sachdev-Ye-Kitaev (SUSY SYK) models. We construct a bi-local superspace, and formulate the bi-local collective superfield theory of the one-dimensional SUSY vector model. The bi-local collective theory provides systematic analysis of the SUSY SYK models. We find that this bi-local collective theory naturally leads to supermatrix formulation in the bi-local superspace. This supermatrix formulation drastically simplifies the analysis of the SUSY SYK models. We also study $\mathcal{N}=1$ bi-local superconformal generators in the supermatrix formulation, and find the eigenvectors of teh superconformal Casimir. We diagonalize the quadratic action in large $N$ expansion.