Charged structure constants from modularity

Abstract

Abstract We derive a universal formula for the average heavy-heavy-light structure constants for 2 d CFTs with non-vanishing $$ \mathfrak{u}(1) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>u</mml:mi> <mml:mfenced> <mml:mn>1</mml:mn> </mml:mfenced> </mml:math> charge. The derivation utilizes the modular properties of one-point functions on the torus. Refinements in $$ \mathcal{N}=2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:math> 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 $$ \mathfrak{u}(1) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>u</mml:mi> <mml:mfenced> <mml:mn>1</mml:mn> </mml:mfenced> </mml:math> symmetry [1]. 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 a useful probe of detecting black hole hair.