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

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.