Modular crossings, OPE coefficients and black holes

TL;DR

The paper exploits the pillow geometry to recast crossing symmetry in 2D CFTs as a modular constraint, enabling a universal analysis of OPE data when one operator is averaged over heavy primaries. By deriving the heavy-state asymptotics via the modular properties and the Cardy density, it shows that the mean-squared heavy OPE coefficients are exponentially suppressed, $\overline{f^2_{\mathcal{O}\mathcal{O}\Delta}} \sim e^{-S_{\rm BH}/2}$, with $S_{\rm BH}$ identified as the BTZ black hole entropy in the holographic dual. In holographic CFTs with large $c$, the averaged heavy channel corresponds to black-hole exchange in the bulk, reproducing the gravitational 2-to-2 S-matrix suppression expected for black-hole intermediate states. These results link modular bootstrap to black-hole physics, imply OPE convergence in Virasoro CFTs with $c>1$, and suggest avenues for extracting broader statistics (e.g., off-diagonal and higher-genus cases) via the pillow-map approach.

Abstract

In (1+1)-d CFTs, the 4-point function on the plane can be mapped to the pillow geometry and thereby crossing symmetry gets translated into a modular property. We use these modular features to derive a universal asymptotic formula for OPE coefficients in which one of the operators is averaged over heavy primaries. The coarse-grained heavy channel then reproduces features of the gravitational 2-to-2 S-matrix which has black holes as their intermediate states.

Figures (1)

• Figure 1: The pillow geometry $\mathbb{P}^1$ as a quotient of torus ($\mathbb{T}^2/\mathbb{Z}_2$). The CFT is quantized on the A-cycle (length $2\pi$) and propagation is along the B-cycle (which is halved compared to that of the original torus). The 4 dots are the fixed points of the orbifold at which the primaries are located. Crossing symmetry implies exchanging the A and B-cycles.