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

Hecke Relations among 2d Fermionic RCFTs