A Cardy Formula for Three-Point Coefficients: How the Black Hole Got its Spots

TL;DR

This paper derives an asymptotic formula for three-point coefficients in 2D CFTs using modular covariance, linking black hole microstates to conformal data and supporting the emergence of BTZ black hole geometry.

Contribution

It generalizes Cardy's formula to three-point coefficients and connects conformal field theory results with black hole physics in AdS_3.

Findings

Asymptotic formula for light-heavy-heavy three-point coefficients

Matching of CFT results with AdS_3 black hole computations

Evidence for emergence of BTZ black hole geometry from microstates

Abstract

Modular covariance of torus one-point functions constrains the three point function coefficients of a two dimensional CFT. This leads to an asymptotic formula for the average value of light-heavy-heavy three point coefficients, generalizing Cardy's formula for the high energy density of states. The derivation uses certain asymptotic properties of one-point conformal blocks on the torus. Our asymptotic formula matches a dual AdS_3 computation of one point functions in a black hole background. This is evidence that the BTZ black hole geometry emerges upon course-graining over a suitable family of heavy microstates.