Fermionic rational conformal field theories and modular linear differential equations

Abstract

Abstract We define modular linear differential equations (MLDE) for the level-two congruence subgroups $\Gamma_\theta$, $\Gamma^0(2)$ and $\Gamma_0(2)$ of $\text{SL}_2(\mathbb Z)$. 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 holomorphic MLDEs without 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.