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.

23 pages, 1 Figure

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• Universal Spectrum of 2d Conformal Field Theory in the Large c Limit
• Poincare Series, 3D Gravity and CFT Spectroscopy
• Holographic interpretation of 1-point toroidal block in the semiclassical limit
• Small Black Holes and Near-Extremal CFTs

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