AdS3/CFT2 on a 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 a finite temperature tend to the standard CFT correlation functions in the limit of removed regularization. In the sum over geometries in both the regular and the $\mathbb Z_N$ orbifold cases, the two-point correlation function for massless modes transforms into a finite sum of products of the conformal-anticonformal CFT Green's functions up to divergent terms proportional to the volume of the $SL(2,\mathbb Z)/\mathbb Z$ group.