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ETH and Modular Invariance of 2D CFTs

Yasuaki Hikida, Yuya Kusuki, Tadashi Takayanagi

TL;DR

This work shows that the modular invariance of torus two-point functions in 2D CFTs imposes ETH-like constraints on heavy-light-heavy three-point functions, yielding exponential suppression of off-diagonal matrix elements controlled by the entropy $S(E)=4\pi\sqrt{cE/12}$. It provides complementary, bootstrap-based support and extends the analysis to multi-point torus correlators, where averages of products of three-point functions scale as $e^{(1-N)S(E)}$, in line with random-matrix expectations. Parallelly, the authors develop open-closed duality constraints for boundary CFTs, deriving Cardy-like growth for open-string densities and detailed bounds on disk one-point coefficients, with a holographic interpretation via BTZ black holes and geodesic worldlines. Overall, the results reinforce the view that holographic 2D CFTs exhibit chaotic, ETH-compatible behavior and establish universal relationships between high-energy CFT data and boundary-state structures, suggesting avenues for exploring randomness in the ETH matrix $R_{nm}$ and concrete solvable boundary-state examples.

Abstract

We study properties of heavy-light-heavy three-point functions in two-dimensional CFTs by using the modular invariance of two-point functions on a torus. We show that our result is non-trivially consistent with the condition of ETH (Eigenstate Thermalization Hypothesis). We also study the open-closed duality of cylinder amplitudes and derive behaviors of disk one-point functions.

ETH and Modular Invariance of 2D CFTs

TL;DR

This work shows that the modular invariance of torus two-point functions in 2D CFTs imposes ETH-like constraints on heavy-light-heavy three-point functions, yielding exponential suppression of off-diagonal matrix elements controlled by the entropy . It provides complementary, bootstrap-based support and extends the analysis to multi-point torus correlators, where averages of products of three-point functions scale as , in line with random-matrix expectations. Parallelly, the authors develop open-closed duality constraints for boundary CFTs, deriving Cardy-like growth for open-string densities and detailed bounds on disk one-point coefficients, with a holographic interpretation via BTZ black holes and geodesic worldlines. Overall, the results reinforce the view that holographic 2D CFTs exhibit chaotic, ETH-compatible behavior and establish universal relationships between high-energy CFT data and boundary-state structures, suggesting avenues for exploring randomness in the ETH matrix and concrete solvable boundary-state examples.

Abstract

We study properties of heavy-light-heavy three-point functions in two-dimensional CFTs by using the modular invariance of two-point functions on a torus. We show that our result is non-trivially consistent with the condition of ETH (Eigenstate Thermalization Hypothesis). We also study the open-closed duality of cylinder amplitudes and derive behaviors of disk one-point functions.

Paper Structure

This paper contains 14 sections, 53 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Modular Invariance on Torus and ETH
  3. Constraints on 3pt Functions
  4. H-L-L 3pt Functions from Conformal Bootstrap
  5. Holographic CFTs
  6. Comparison with ETH
  7. Torus Multi-Point Functions and Modular Invariance
  8. Constraints from Open-Closed Duality
  9. Duality for Vacuum Cylinder Amplitudes
  10. $t\to 0$ Limit
  11. $s\to 0$ Limit
  12. Holographic Interpretation
  13. Duality for 1pt Function on Cylinder
  14. Conclusions

Figures (1)

  • Figure 1: (a) A closed string state at high energy and a light open string state are approximately described by the BTZ black hole and a perturbative scalar particle, respectively. (b) A bulk light particle dual to a boundary open string state can be regarded as a low energy excitation of a bulk open string on a brane.