# Witten Diagrams for Torus Conformal Blocks

## TL;DR

This paper provides a holographic framework for understanding conformal blocks in 2D CFTs on the sphere and torus, using Witten diagrams and Wilson line networks in AdS and Chern-Simons gravity.

## Contribution

It introduces a novel holographic description of torus conformal blocks via Witten diagrams and Wilson line networks, connecting CFT calculations with bulk gravity models.

## Key findings

- Conformal blocks on the torus can be expressed as Witten diagrams in thermal AdS.
- The semi-classical limit of n-point blocks corresponds to bulk world-line actions.
- Equivalence between Witten diagram and Wilson line formulations is established.

## Abstract

We give a holographic description of global conformal blocks in two dimensional conformal field theory on the sphere and on the torus. We show that the conformal blocks for one-point functions on the torus can be written as Witten diagrams in thermal AdS. This is accomplished by deriving a general conformal Casimir equation for global conformal blocks, and showing that Witten diagrams obey the same equation. We study the semi-classical limit of n-point conformal blocks, and show that these equal the action of a network of bulk world-lines obeying appropriate geodesic equations. We give an alternate description in the Chern-Simons formulation of 3D gravity, where the conformal blocks are described by networks of Wilson lines, and argue that these formulations are equivalent.