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

Per Kraus, Alexander Maloney

TL;DR

The paper develops a Cardy-like framework for three-point coefficients in 2D CFTs by exploiting modular covariance of torus one-point functions, linking high-energy heavy-state data to light-sector information. It derives an explicit asymptotic formula for the average light-heavy-heavy OPE coefficient and validates it via a bulk AdS3 calculation in the BTZ black hole background, suggesting black hole geometry emerges upon coarse-graining over heavy microstates. The authors extend the analysis to primary operators using torus blocks, uncovering a universal structure with a shift c→c−1 attributed to Virasoro descendants. This work tightens the CFT/AdS3 correspondence by connecting modular constraints to bulk geometric interpretations and providing concrete asymptotics for fundamental CFT data beyond the spectrum. Overall, it highlights how gravitational features of BTZ black holes manifest in averaged CFT observables and informs future precision studies of three-point data in large-c theories.

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.

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

TL;DR

The paper develops a Cardy-like framework for three-point coefficients in 2D CFTs by exploiting modular covariance of torus one-point functions, linking high-energy heavy-state data to light-sector information. It derives an explicit asymptotic formula for the average light-heavy-heavy OPE coefficient and validates it via a bulk AdS3 calculation in the BTZ black hole background, suggesting black hole geometry emerges upon coarse-graining over heavy microstates. The authors extend the analysis to primary operators using torus blocks, uncovering a universal structure with a shift c→c−1 attributed to Virasoro descendants. This work tightens the CFT/AdS3 correspondence by connecting modular constraints to bulk geometric interpretations and providing concrete asymptotics for fundamental CFT data beyond the spectrum. Overall, it highlights how gravitational features of BTZ black holes manifest in averaged CFT observables and informs future precision studies of three-point data in large-c theories.

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.

Paper Structure

This paper contains 15 sections, 82 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A Cardy formula for three-point coefficients
  3. Torus one-point functions
  4. Modular invariance
  5. The asymptotic formula
  6. The AdS$_3$ interpretation
  7. The AdS${}_3$ setup
  8. Witten diagram calculation
  9. Three-point function coefficients for primary operators
  10. Torus blocks
  11. Asymptotics of three-point coefficients
  12. Asymptotics of torus conformal blocks
  13. Setup
  14. Large $\Delta$ limit
  15. Validity of large $\Delta$ asymptotics

Figures (1)

  • Figure 1: Geodesic approximation to the one-loop contribution to the one-point function of $\phi_O$. A constant time slice of the BTZ metric is depicted. The $\phi_\chi$ particle wraps the horizon and meets the $\phi_O$ particle emanating from the boundary at a cubic vertex. If the geodesic worldlines are replaced by propagators and the cubic vertex is integrated over spacetime, this becomes a full-fledged one-loop Witten diagram.