Holographic description of 2D conformal block in semi-classical limit

In this paper, we study the holographic descriptions of the conformal block of heavy operators in two-dimensional large c conformal field theory. We consider the case that the operators are pairwise inserted such that the distance between the operators in a pair is much smaller than the others. In this case, each pair of heavy operators creates a conical defect in the bulk. We propose that the conformal block is dual to the on-shell action of three dimensional geometry with conical defects in the semi-classical limit. We show that the variation of the on-shell action with respect to the conical angle is equal to the length of the corresponding conical defect. We derive this differential relation on the conformal block in the field theory by introducing two extra light operators as both the probe and the perturbation. Our study also suggests that the area law of the holographic Rényi entropy must holds for a large class of states generated by a finite number of heavy operators insertion.