# Structure Constants from Modularity in Warped CFT

## TL;DR

This paper derives a universal formula for the asymptotic growth of three-point coefficients in Warped CFTs and confirms it through holographic calculations involving BTZ black holes, highlighting the emergence of black hole geometries from microstates.

## Contribution

It provides the first universal asymptotic formula for three-point coefficients in WCFTs and demonstrates its consistency with holographic BTZ black hole calculations.

## Key findings

- Universal asymptotic growth formula for WCFT three-point coefficients
- Agreement between WCFT and holographic BTZ black hole calculations
- Evidence for black hole geometry emergence from WCFT microstates

## Abstract

We derive a universal formula for the asymptotic growth of the mean value of three-point coefficient for Warped Conformal Field Theories (WCFTs), and provide a holographic calculation in BTZ black holes. WCFTs are two dimensional quantum field theories featuring a chiral Virasoro and U(1) Kac-Moody algebra, and are conjectured to be holographically dual to quantum gravity on asymptotically AdS$_3$ spacetime with Comp$\grave{\mathrm{e}}$re-Song-Strominger boundary conditions. The WCFT calculation amounts to the calculation of one-point functions on torus, whose high temperature limit can be approximated by using modular covariance of WCFT, similar to the derivation of Cardy formula. The bulk process is given by a tadpole diagram, with a massive spinning particle propagates from the infinity to the horizon, and splits into particle and antiparticle which annihilate after going around the horizon of BTZ black holes. The agreement between the bulk and WCFT calculations indicates that the black hole geometries in asymptotically AdS$_3$ spacetimes can emerge upon coarse-graining over microstates in WCFTs, similar to the results of Kraus and Maloney in the context of AdS/CFT[arXiv:1608.03284].