SYK Model and Its Generalization

Abstract

We briefly review the Sachdev-Ye-Kitaev (SYK) model and study several generalizations of it. First, we consider an SYK model with global U(1) charge and find its four-point function in Euclidean and real time. Then, we proceed to add N=2 supersymmetry. This N=2 SYK model is supposed to be dual to the near-horizon geometry of a stable black hole in four-dimensional supergravity. We study this model in one and two dimensions, find the four-point functions and corresponding chaos exponents. Next, we briefly review tensor models showing SYK-like behavior. We study operators unique to the tensor model (compared to the SYK) and count them using the partition function of large N gauge theory. Finally, we switch to the (approximate) gravity dual of the SYK, the Jackiw-Teitelboim theory of dilaton gravity. We study correlators of heavy operators on the boundary of the latter theory using the holographic prescription. We find some novel properties of these correlators, such as having a finite limit at large Euclidean distances. We are also able to study out-of-time ordered four-point function and find that it approaches an exponentially small limit at late times.