Fermionic rational conformal field theories and modular linear differential equations

Abstract

We define modular linear differential equations (MLDE) for the level-two congruence subgroups Γθ, Γ0(2) and Γ0(2) of SL2(Z). Each subgroup corresponds to one of the spin structures on the torus. The pole structures of the fermionic MLDEs are investigated by exploiting the valence formula for the level-two congruence subgroups.We focus on the first- and second-order holomorphicMLDEswithout poles and use them to find a large class of "fermionic rational conformal field theories"(fermionic RCFTs), which have non-negative integer coefficients in the q-series expansion of their characters.We study the detailed properties of these fermionic RCFTs, some of which are supersymmetric. This work also provides a starting point for the classification of the fermionic modular tensor category.