Perturbative classical conformal blocks as Steiner trees on the hyperbolic disk

TL;DR

This paper explores the connection between holographic Steiner trees in hyperbolic geometry and large-$c$ conformal blocks in the AdS/CFT correspondence, providing explicit examples and analyzing their geometric and CFT properties.

Contribution

It introduces a class of holographic Steiner trees inscribed in polygons that are dual to large-$c$ conformal blocks, with explicit calculations and geometric analysis.

Findings

Holographic Steiner trees can be inscribed in polygons with ideal vertices.

Connectivity and cuts of Steiner trees encode conformal block factorization.

Explicit examples for N=2,3,4 Steiner trees are provided.

Abstract

We consider the Steiner tree problem in hyperbolic geometry in the context of the AdS/CFT duality between large-$c$ conformal blocks on the boundary and particle motions in the bulk. The Steiner trees are weighted graphs on the Poincare disk with a number of endpoints and trivalent vertices connected to each other by edges in such a way that an overall length is minimum. We specify a particular class of Steiner trees that we call holographic. Their characteristic property is that a holographic Steiner tree with $N$ endpoints can be inscribed into an $N$-gon with $N-1$ ideal vertices. The holographic Steiner trees are dual to large-$c$ conformal blocks. Particular examples of $N=2,3,4$ Steiner trees as well as their dual conformal blocks are explicitly calculated. We discuss geometric properties of the holographic Steiner trees and their realization in CFT terms. It is shown that connectivity and cuts of the Steiner trees encode the factorization properties of large-$c$ conformal blocks.