$\mathcal N=2$ SYK model in the superspace formalism

TL;DR

The paper develops a superspace framework for the $ abla$N=2 SYK model in one dimension and its two-dimensional generalization, solving for the conformal four-point function by expanding in the eigenfunctions of the $su(1,1|1)$ Casimir and analyzing the SYK kernel's action. The authors construct the shadow representation, compute explicit kernel eigenvalues in both antisymmetric and symmetric channels, and establish the (non)Hilbert space structure of the eigenfunctions, including the intriguing $h=0$ and $h=-1$ modes. They extend the analysis to two dimensions, derive the two-dimensional kernel and its spectrum, and extract the central charge $c=3N(1-2Δ)$, while showing that the retarded kernel in 2D is non-maximally chaotic. Overall, the work provides a detailed, technically rich map from superconformal Casimir eigenfunctions to the full four-point function and chaos properties in $ abla$N=2 SYK.

Abstract

We use superspace methods to study an SYK-like model with $\mathcal N=2$ supersymmetry in one dimension, and an analog of this model in two dimensions. We find the four-point function as an expansion in the basis of eigenfunctions of the Casimir of $su(1,1|1)$. We also find retarded kernels and Lyapunov exponents for both cases.

Figures (8)

• Figure 1: Schwinger--Dyson equation for the two-point function. The melonic part contains an even number of propagators.
• Figure 2: $\mathcal{N}=2$ conformal kernel.
• Figure 3: Eigenvalues of the antisymmetric (red) and symmetric (blue) kernels at $\hat{q}=5$.
• Figure 4: Zero--rung four-point function.
• Figure 5: The integration contour for the $\mathcal{N}=2$ SYK model avoids the double pole at zero.