Notes on complex $q=2$ SYK

This note clarifies and extends results on the complex SYK model to the solvable q = 2 case. We calculate the four point function OPE of fermions in the low energy CFT, implying the existence of a tower of integer-weight operators in the IR. We comment on the lack of a mode breaking conformal symmetry in this special case of SYK and the consequences for deformations of the theory near the conformal fixed point. We use the nearly-free structure of the model to provide a closed form expression for OPE coefficients of the integer-weight operators. We also discuss analytic and numerical results relevant to the thermodynamics of q = 2 SYK in both the complex and real case. The tower of operators transform in the discrete series of representations of SL(2,R), the representations shared by dS2 and AdS2. In this work we continue discussion of holographic models including these representations.