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

TL;DR

This work proves that in the heavy-light semiclassical limit, $n$-point classical conformal blocks of a 2d CFT are equivalent to the lengths of dual geodesic networks on the angle-deficit $AdS_{3}$ space for any $n$. The authors formulate both the boundary (monodromy/accessory-parameter) problem and the bulk (geodesic network) problem as potential vector-field systems and show their governing algebraic equations share the same roots through a weak equivalence argument, guaranteeing the correspondence without requiring explicit block or length formulas. A heavy-light perturbation framework is developed to relate boundary accessory parameters to bulk angular momenta and vertex/angular constraints, enabling explicit matching of structures via vertex equilibrium, radial positions, and angular balance. The result generalizes known $4$-point relations to arbitrary $n$ and provides a robust foundation for exploring higher-genus generalizations and $1/c$ corrections while clarifying the role of extra bulk roots not mapped to boundary data.

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.

Figures (5)

• Figure 1: The $(n+1)$-point conformal block in a particular channel, where the degenerate operator fuses with $\mathcal{O}_m$ to yield $\mathcal{O}_{m+1}$ for any $m = 1,...\,, n-3$. There are $2(n-3)$ blocks of this type arising as solutions of the decoupling condition. In the limit $c\to\infty$ the degenerate operator decouples and therefore the only surviving block is given by that one shown on Fig. \ref{['block']}. We fix $z_1=0$, $z_{n-1} = 1$, $z_n = \infty$.
• Figure 2: The $n$-point conformal block. Two bold black lines are background heavy operators, thin blue lines represent primary and exchanged perturbative heavy operators which are discussed in Section \ref{['sec:pert']}.
• Figure 3: Network of geodesic lines on the hyperbolic disk. Solid and wave lines denote respectively external ($\epsilon_m$) and exchanged ($\tilde{\epsilon}_k$) particles, dotted lines denote the middle part of the graph. The boundary attachment points are $w_m$, $m=1,...\,,n-2$.
• Figure 4: Cubic vertex on the hyperbolic disk. Incoming ($I$, $K$) and outcoming ($J$) momenta are constrained by the equilibrium condition. The radial vertex position is parameterized by $\eta = \cot^2 \rho$, where $\rho$ is the radial distance from the center.
• Figure 5: Angular separations. Dotted lines show angular positions $w_i$ and $\psi_i$ of the boundary attachment points and vertices, respectively.

Theorems & Definitions (8)

• Proposition 4.1
• Proposition 5.1
• Definition 5.2
• Proposition 5.3
• Lemma 5.4
• Lemma 5.5
• Proposition 5.6
• Proposition 5.7