# Four-point conformal blocks with three heavy background operators

## TL;DR

This paper computes four-point conformal blocks in 2D CFT with three heavy background operators and one light operator using the monodromy method at large central charge, linking them to geodesic lengths in a holographic AdS3 space with conical singularities.

## Contribution

It introduces a method to calculate conformal blocks with multiple heavy backgrounds and a light operator, extending the block/length correspondence to higher-point functions.

## Key findings

- Conformal blocks are computed via the monodromy method in the large central charge limit.
- Background operators create an AdS3 space with three conical singularities.
- Geodesic lengths in the bulk correspond to perturbative conformal blocks.

## Abstract

We study CFT$_2$ Virasoro conformal blocks of the 4-point correlation function $\langle \mathcal{O}_L \mathcal{O}_H \mathcal{O}_H \mathcal{O}_H \rangle $ with three background operators $\mathcal{O}_H$ and one perturbative operator $\mathcal{O}_L$ of dimensions $\Delta_L/\Delta_H \ll1$. The conformal block function is calculated in the large central charge limit using the monodromy method. From the holographic perspective, the background operators create $AdS_3$ space with three conical singularities parameterized by dimensions $\Delta_H$, while the perturbative operator corresponds to the geodesic line stretched from the boundary to the bulk. The geodesic length calculates the perturbative conformal block. We propose how to address the block/length correspondence problem in the general case of higher-point correlation functions $\langle \mathcal{O}_L \cdots \mathcal{O}_L \mathcal{O}_H \cdots \mathcal{O}_H \rangle $ with arbitrary numbers of background and perturbative operators.