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

TL;DR

The paper develops a bi-local collective superfield theory for one-dimensional $ ext{N}=1,2$ SUSY vector models, with a focus on SUSY SYK models. It shows that constructing a bi-local superspace yields a natural supermatrix representation, drastically simplifying large-$N$ analyses and the diagonalization of the quadratic action. For $ ext{N}=1$, it derives bi-local superconformal generators, solves the Casimir eigenproblem, and diagonalizes the quadratic action; it also extends the framework to $ ext{N}=2$ with chiral/anti-chiral bi-local sectors. These results provide a unified, tractable method to study SUSY vector models and lay groundwork for higher $ ext{N}$ and higher-dimensional generalizations, with potential impact on AdS/CFT and quantum chaos in SUSY contexts.

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.