Perturbative classical conformal blocks as Steiner trees on the hyperbolic disk

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.

40 pages, 13 figures; v2: misprints corrected, a new Section 3.4 on Mobius transformations added

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• Holographic variables for CFT${}_2$ conformal blocks with heavy operators
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• More on Wilson toroidal networks and torus blocks
• Four-point conformal blocks with three heavy background operators
• Semiclassical torus blocks in the t-channel
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• Classical conformal blocks, Coulomb gas integrals and Richardson-Gaudin models
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• Reparametrization mode Ward Identities and chaos in higher-pt. correlators in CFT${}_2$
• Rate of cluster decomposition via Fermat-Steiner point
• Differential equations for classical Virasoro blocks with heavy and light operators
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