Semiclassical OPE coefficients from 3D gravity

TL;DR

This work derives a universal semiclassical formula for OPE coefficients in 2D CFTs with a weak light spectrum by equating the crossing equation to the fusion transformation of the vacuum Virasoro block and evaluating the semiclassical fusion kernel. The authors express the universal OPE coefficients as the semiclassical limit of the fusion kernel, and they corroborate this with a gravity calculation: the regularized Einstein-Hilbert action for a geometry with three conical defects, analytically continued, reproduces the same universal result after Liouville normalization. A key insight is that when deficit angles violate the triangle inequality, the bulk geometry develops a horizon-like structure, hinting at a deeper link between universality in CFT data and bulk causal structure. The study thus unifies a conformal bootstrap derivation with a holographic bulk computation, using Liouville theory and 3D gravity to illuminate the interplay between crossing symmetry, OPE data, and gravitational saddles. Potential extensions include higher-dimension operators, BTZ regimes, and more intricate bulk topologies such as multi-boundary wormholes.

Abstract

We present a closed form expression for the semiclassical OPE coefficients that are universal for all 2D CFTs with a "weak" light spectrum, by taking the semiclassical limit of the fusion kernel. We match this with a properly regularized and normalized bulk action evaluated on a geometry with three conical defects, analytically continued in the deficit angles beyond the range for which a metric with positive signature exists. The analytically continued geometry has a codimension-one coordinate singularity surrounding the heaviest conical defect. This singularity becomes a horizon after Wick rotating to Lorentzian signature, suggesting a connection between universality and the existence of a horizon.

Figures (5)

• Figure 1: Three conical defects joining in 3D hyperbolic space.
• Figure 2: The contour $\mathbb T$ in the definition of the fusion kernel. When $\eta_{ext}$ and $\eta_s$ are real, small and positive regulators are turned on so that the arrays of poles do not overlap, see \ref{['Regulators']}. This figure is drawn with the choice $2\epsilon_{ext} : \epsilon_s : \text{Im}\,\eta_t = 3 : 1 : 5$. The solid dot is the dominant critical point $s_-$.
• Figure 3: The contour $\mathbb{S}$ when $\eta_{ext} \leq {1\over4}$.
• Figure 4: Regimes of validity of the heavy particle worldline computation and the conformal bootstrap analysis.
• Figure 5: Left: Two heavy particles (double-line) joining with a conical defect (zigzag), when the triangle inequality is violated. The geometry has positive signature in the radial direction, but negative signature in the angular directions. The cone depicts a coordinate singularity. Right: After Wick rotating to Lorentzian signature, the Penrose diagram for the creation of a conical defect by two heavy particles. Each point on this diagram away from $\rho = 0$ represents a circle, and the two particles come in from $\theta = 0, \pi$. The coordinate singularity becomes a horizon at $\rho = 1$. The geometry near the horizon in patch III is an FLRW universe \ref{['FLRW']}, which does not see the singularity at $\rho = 0$.