Mini-course on some aspects of the SYK model

The Sachdev-Ye-Kitaev (SYK) model is a quantum mechanical theory with N fermions with p-body interactions and random coupling constants. In the large N limit it can be solved exactly and shown to be dual to a gravitational theory and to be maximally chaotic. In the first part of this mini-course we will review in detail the solution of the model in the so-called double scaling limit, in which both p and N are large but p^2/N is fixed. The partition function and the correlators of the model will be computed exactly and the connection with quantum gravity will be uncover. The second part of the course will focus on the chaotic behavior of the SYK model. We will compute out-of-time ordered correlators (OTOCs), which in a chaotic system grow exponentially. The rate of growth of the OTOCs is controlled by the Lyapunov exponent, which we will be able to compute explicitly in a certain limit and see that the system is maximally chaotic (i.e., it saturates a previously conjectured bound on chaos).