More on Supersymmetric and 2d Analogs of the SYK Model

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.

• The paper introduces supersymmetric and 2D SYK extensions by constructing superconformal eigenfunctions to analyze four-point functions.
• The paper derives a novel integral formula enabling precise computations of the central charge and chaos exponent in the IR limit.
• The paper identifies challenges in extending global symmetries in 2D models, highlighting divergence issues and implications for holography.

Analysis of Supersymmetric and Two-Dimensional Analogs of the SYK Model

The paper "More On Supersymmetric And 2d Analogs of the SYK Model" investigates theoretical extensions of the Sachdev-Ye-Kitaev (SYK) model. These extensions incorporate supersymmetric elements and adaptations to two-dimensional fields, extending the analytical toolkit developed for the original SYK model.

The authors commence by constructing a basis of superconformal eigenfunctions for the expansion of a four-point function, facilitating a detailed analysis of the one-dimensional supersymmetric SYK model. They proceed to explore two-dimensional bosonic and supersymmetric variants of the SYK model, which assume interactions between $N$ fields governed by $q$-field couplings subject to randomization. Notably, the purely bosonic model exhibits problematic behavior, prompting the authors to focus on a promising supersymmetric model.

In this supersymmetric model, contrary to pure bosonic SISYK analogs, the authors demonstrate the emergence of a large $N$ conformal field theory (CFT) in the infrared (IR) limit. Particularly, the model sustains the interaction strength at order one, which is critical to establishing conformality in the IR spectrum. A novel integral formula is derived for the four-point function at the leading $1/N$ order, allowing the computation of central charge, chaos exponent, and anomalous dimensions.

The authors identify a significant issue while attempting to replicate an SYK-like conformal field theory (CFT) with a continuous global symmetry in two dimensions; such models complicate owing to divergences associated with would-be symmetry currents. Speculating on the analytical structure, the paper suggests the obstructions stem from the fulfillment of conformal symmetry, which raises fundamental questions about the nature of CFTs in higher-dimensional SYK analogs.

Furthermore, the analysis calculates the eigenvalues of the ladder kernel in these models, explaining their relevance in establishing the chaos exponent's bounded rate ($\lambda_L \approx 0.5824$ for $q = 3$), revealing an exponential growth below the theoretical maximum consistent with chaos, the so-called "chaos bound."

Key Findings and Implications

1. Ladder Kernel and IR Conformal Invariance: The authors provide a comprehensive computation of the ladder diagram contributions to the four-point correlation functions, observing their role in stabilizing IR conformal invariance in supersymmetric extensions.
2. Analytical Continuation: The paper shows how analytical continuation of eigenvalues and eigenfunctions may reveal subtle features of the model’s response to high-energy fluctuations, linked to chaos and non-normalizability in a two-dimensional holographic framework.
3. Central Charge Computation: A noteworthy computational result is the derivation of the central charge in the supersymmetric model, which responds consistently to varying $q$, corroborating the conformal nature in the large $N$ limit.
4. Challenges in Extending Global Symmetries: The convergence difficulties in formulating global symmetries within two-dimensional analogs emphasize the distinction between algebraic convenience and physical realizability of higher-dimensional SYK-like models.
5. Extensions and Novel CFTs: Proposed is a frame of analyzing CFTs that emerge from strongly interacting, high-dimensional SUK variants. While the paper primarily restrains itself to analysis without an overt gravitational dual, its results remain instrumental in mapping the landscape of strongly correlated fields.

Consequently, this paper emphasizes the theoretical richness of supersymmetric and two-dimensional SYK analogs and their dimensional families, encouraging future studies to resolve identified issues with potential further insights into quantum field dynamics and holographic correspondence. The treatment offers substantial calculations and argumentation influential in the ongoing discussions of (super)conformal fields and holography, shifting towards even greater dimensional frameworks.