Holographic Description of 2D Conformal Block in Semi-classical Limit

TL;DR

This paper explores the holographic dual of 2D conformal blocks with heavy operators, revealing a geometric interpretation involving conical defects and deriving key relations between the on-shell action and defect lengths.

Contribution

It introduces a holographic description of conformal blocks with conical defects and establishes a differential relation linking the on-shell action to defect geometry in the semi-classical limit.

Findings

Conformal blocks correspond to on-shell actions of 3D geometries with conical defects.

Variation of the on-shell action relates to the length of conical defects.

The area law for holographic Renyi entropy applies to states with heavy operator insertions.

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.