We study the SYK model in the large N limit beyond the replica-diagonal approximation. First we show that there are exact replica-nondiagonal solutions of the saddle point equations for q = 2 for any finite replica number M . In the interacting q = 4 case we are able to construct the numerical solutions, which are in one-to-one correspondence to the analytic solutions of the quadratic model. These solutions are singular in the M → 0 limit in both quadratic and quartic interaction cases. The calculations of the on-shell action at finite integer M show that the nondiagonal replicasymmetric saddles are subleading in both quadratic and quartic cases.We also study replica-nondiagonal solutions of the SYK in the strong coupling limit. For arbitrary q we show that besides the usual solutions of the replica-diagonal saddle point equations in the conformal limit, there are also replica-nondiagonal solutions for any value of M (including zero). The specific configurations that we study, have factorized time and replica dependencies. The corresponding saddle point equations are separable at strong coupling, and can be solved using the Parisi ansatz from spin glass theory. We construct the solutions which correspond to the replica-symmetric case and to one-step replica symmetry breaking. We compute the regularizized free energy on these solutions in the limit of zero replicas. It is observed that there are nondiagonal solutions with the regularized free energy lower than that of the standard diagonal conformal solution.arXiv:1811.04831v4 [hep-th]

Abstract:We study correlation functions of scalar operators on the boundary of the AdS 3 space deformed by moving massive particles in the context of the AdS/CFT duality. To calculate two-point correlation functions we use the geodesic approximation and the renormalized image method. We compare results of the renormalized image method with direct calculations using tracing of winding geodesics around the cone singularities, and show on examples that they are equivalent. We demonstrate that in the geodesic approximation the correlators exhibit a zone structure. This structure substantially depends on the mass and velocity of the particle.