Many-point classical conformal blocks and geodesic networks on the hyperbolic plane

TL;DR

This paper establishes a holographic duality between n-point conformal blocks in 2d CFT and geodesic networks in AdS3, revealing a geometric interpretation of conformal blocks via particle worldlines.

Contribution

It demonstrates that classical n-point conformal blocks correspond to lengths of geodesic networks, extending the holographic correspondence to multiple points with a new geometric perspective.

Findings

Classical conformal blocks equal geodesic network lengths in AdS3.

Accessory parameters and particle momenta satisfy the same algebraic equations.

The boundary and bulk systems are reformulated as potential vector fields.

Abstract

We study the semiclassical holographic correspondence between 2d CFT n-point conformal blocks and massive particle configurations in the asymptotically AdS3 space. On the boundary we use the heavy-light approximation in which case two of primary operators are the background for the other (n-2) operators considered as fluctuations. In the bulk the particle dynamics can be reduced to the hyperbolic time slice. Although lacking exact solutions we nevertheless show that for any n the classical n-point conformal block is equal to the length of the dual geodesic network connecting n-3 cubic vertices of worldline segments. To this end, both the bulk and boundary systems are reformulated as potential vector fields. Gradients of the conformal block and geodesic length are given respectively by accessory parameters of the monodromy problem and particle momenta of the on-shell worldline action represented as a function of insertion points. We show that the accessory parameters and particle momenta are constrained by two different algebraic equation systems which nevertheless have the same roots thereby guaranteeing the correspondence.