Gauging the complex SYK model

Motivated by SYK-like models describing near-BPS black holes in string/M-theory, we consider gauging the U$(1)$ symmetry of the complex SYK model in the presence of a Wilson line with charge $k$. At a fixed background gauge field, solutions to the Schwinger-Dyson equations display vastly different properties from those at a fixed real chemical potential. In the partition function and the two-point function, the integral over the gauge field is performed either directly or via a large $N$ saddle point approximation, and both results are consistent with exact diagonalization data. From the behaviour of the two-point function at large $N$, we deduce that the conformal symmetry at low energies is preserved at fixed $\kappa = k/N = 0$, but broken at $\kappa\neq 0$. In addition, we find that there is maximal chaos for all $k$.