# Charged structure constants from modularity

## TL;DR

This paper derives a universal formula for average heavy-heavy-light structure constants in 2d CFTs with U(1) charge, connecting modular properties, holography, and black hole hair detection.

## Contribution

It generalizes previous results to include U(1) charges, providing a Cardy-like formula and holographic validation for 2d CFT structure constants.

## Key findings

- Universal formula for charged structure constants derived
- Matches with holographic computations at large central charge
- Provides a probe for black hole hair in holography

## Abstract

We derive a universal formula for the average heavy-heavy-light structure constants for 2d CFTs with non-vanishing u(1) charge. The derivation utilizes the modular properties of one-point functions on the torus. Refinements in N=2 SCFTs, show that the resulting Cardy-like formula for the structure constants has precisely the same shifts in the central charge as that of the thermodynamic entropy found earlier. This analysis generalizes the recent results by Kraus and Maloney for CFTs with an additional global u(1) symmetry. Our results at large central charge are also shown to match with computations from the holographic dual, which suggest that the averaged CFT three-point coefficient also serves as an useful probe of detecting black hole hair.