Fermionic Rational Conformal Field Theories and Modular Linear Differential Equations

Abstract

We define Modular Linear Differential Equations (MLDE) for the level-two congruence subgroups ${\Gamma }_{\vartheta }$, ${\Gamma }^0(2)$ 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', 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.

63 pages, 4 figures, 19 tables, references added, minor corrections