AdS_3/CFT_2 on torus in the sum over geometries

We investigate the AdS$_3$/CFT$_2$ correspondence for the Euclidean AdS$_3$ space compactified on a solid torus with the CFT field on the regularizing boundary surface in the bulk. Correlation functions corresponding to the bulk theory at finite temperature tend to the standard CFT correlation functions in the limit of removed regularization. In both regular and $Z_N$ orbifold cases, in the sum over geometries, the two-point correlation function for massless modes factors, up to divergent terms proportional to the volume of the SL(2,Z)/Z} group, into the finite sum of products of the conformal--anticonformal CFT Green's functions.