Holographic Description of 2D Conformal Block in Semi-classical Limit
Bin Chen, Jie-qiang Wu, Jia-ju Zhang
TL;DR
This work proposes a holographic description of 2D large-$c$ conformal blocks for heavy operators inserted in pairwise fashion, showing the leading block equals the on-shell action of AdS$_3$ gravity with conical defects. A key result is a differential relation, $-n_j^2\frac{\partial}{\partial n_j}\log \mathcal{F}=\frac{L_j}{4G}+f_j$, linking the block to the length $L_j$ of a defect homologous to each operator pair, with $L_j$ read from a regulated geodesic using two light probes as observers. The authors provide a field-theory derivation via monodromy and a probe-based regularization, and verify the framework through explicit two-point and four-point examples, aligning gravity and CFT calculations. The findings extend holographic Rényi entropy ideas to general conformal blocks and suggest an area-law-like structure for a wider class of heavy-state configurations in AdS$_3$/CFT$_2$."
Abstract
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 Renyi entropy must holds for a large class of states generated by a finite number of heavy operators insertion.