Scaling limits of complex Sachdev-Ye-Kitaev models and holographic geometry

We compare different limits of the Sachdev-Ye-Kitaev model of $N$ complex fermion with $p$-fermion interactions. First, we compute the fermion Green's function and free energy in the limit of large $N$ followed subsequently by the limit of large $p$. Next, we examine the `double-scaling' limit in which the large $N,p$ limits are taken at fixed $\lambda = p^2/N$. Earlier results on the latter limit are resummed for small $\lambda$, and shown to match our results for the first limit. We also describe the holographic match of our results to two-dimensional Jackiw-Teitelboim gravity with an additional $U(1)$ gauge field.