Chiral algebras of two-dimensional SYK models

We study chiral algebras in the Q¯ -cohomology of two dimensional SYK models with extended supersymmetry. In a special limit discovered in [1], 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 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 [1]. 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 [1] is robust to 1/N corrections.