AdS₃ Gravity And The Complex Sachdev-Ye-Kitaev Model

We first study the connections between the complex SYK model at the low energy limit and warped conformal field theories(WCFT)by comparing their symmetries,corre-lators and the effective action.The low energy limit of the complex SYK model has an emergent time reparametrization and local U(1)symmetry,which is similar to the warped conformal symmetries in WCFT.Moreover,there is a spontaneous and explicit symmetry breaking in the complex SYK model so that it can be understood as a nearly WCFT.We find that they have the same form of correlation functions due to the similar symmetry structure.Moreover,we prove that the effective action of the complex SYK model in the low energy limit can be compared with the Virasoro generator in WCFT and the partition function of the complex SYK model at the low energy matches the vacuum character of warped conformal theories in the semiclassical limit.We then review the Compère-Song-Strominger(CSS)boundary conditions for Ad S₃gravity which is dual to warped confor-mal theories and give a non Lorentzian generalization.The holographic dual determined by the generalized boundary conditions can be understood as an irrelevant deformation of a warped conformal field theory.Upon consistent reduction to two dimensions,Ad S₃gravity leads to a dilaton gravity model coupled to a U(1)gauge field.We show that near extremality the boundary terms inherited from generalized CSS boundary conditions in three dimensions results in an effective action which can be identified with the effective action of the complex Sachdev-Ye-Kitaev models at the low energy limit.Besides the Schwarzian term reflecting the breaking of conformal symmetry,the effective action con-tains an additional anomaly term that captures the breaking of the local U(1) symmetry.