# Chiral Algebras of Two-Dimensional SYK Models

## TL;DR

This paper explores the structure of chiral algebras in two-dimensional supersymmetric SYK models, revealing new higher-spin algebras and their relation to string theory in a special limit, with implications for understanding symmetries beyond criticality.

## Contribution

It explicitly constructs vertical and horizontal single-particle higher-spin algebras in a special limit of 2D SYK models, linking them to stringy structures and analyzing their robustness at finite N.

## Key findings

- Vertical higher-spin algebra is bilinear in fundamental fields.
- Horizontal higher-spin algebra appears in the special limit.
- Absence of single-particle higher-spin algebra away from the limit.

## Abstract

We study chiral algebras in the $\bar{Q}$-cohomology of two dimensional SYK models with extended supersymmetry. In a special limit discovered in arXiv:1805.09325, we are able to construct explicitly a "vertical" single-particle higher-spin algebra that is bilinear in the fundamental fields. This algebra can be regarded as the counterpart, when going away from criticality, of the infrared emergent higher-spin symmetry of the $\mathcal{N}=(0,2)$ SYK model. Moreover, a second "horizontal" single-particle higher-spin algebra appears in this limit. Together with the vertical algebra they generate a stringy algebra with a "higher spin square" structure that is believed to appear in the tensionless limit of string theory. On the other hand, we do not find single-particle higher-spin algebra away from the special limit, which is consistent with the result in arXiv:1805.09325. Our analysis is carried out for each individual realization of the random couplings and for finite $N$ (and $M$), which in particular indicates that the conclusion in arXiv:1805.09325 is robust to $1/N$ corrections.