Hecke Relations among 2d Fermionic RCFTs

TL;DR

This paper introduces a fermionic Hecke operator that connects characters of 2d fermionic RCFTs, revealing that many known supersymmetric theories can be generated as its images, thus uncovering new relations among these theories.

Contribution

The paper generalizes the Hecke operator to fermionic RCFTs, establishing a new mathematical framework for relating different 2d fermionic conformal field theories.

Findings

Almost all known supersymmetric RCFTs are fermionic Hecke images of simple theories.

The fermionic Hecke operator maps NS characters to NS characters and extends to other sectors.

Fermionic Hecke relations help understand coset relations among SCFTs with specific central charges.

Abstract

Recently, Harvey and Wu proposed a suitable Hecke operator for vector-valued $SL(2,\mathbb{Z})$ modular forms to connect the characters of different 2d rational conformal field theories (RCFTs). We generalize such an operator to the 2d fermionic RCFTs and call it fermionic Hecke operator. The new Hecke operator naturally maps the Neveu-Schwarz (NS) characters of a fermionic theory to the NS characters of another fermionic theory. Mathematically, it is the natural Hecke operator on vector-valued $\Gamma_\theta$ modular forms of weight zero. We find it can also be extended to $\mathrm{\widetilde{NS}}$ and Ramond (R) sectors by combining the characters of the two sectors together. We systematically study the fermionic Hecke relations among 2d fermionic RCFTs with up to five NS characters and find that almost all known supersymmetric RCFTs can be realized as fermionic Hecke images of some simple theories such as supersymmetric minimal models. We also study the coset relations between fermionic Hecke images with respect to $c=12k$ holomorphic SCFTs.