On the modular covariance properties of composite fermions on the torus

In this work we show that the composite fermion construction for the torus geometry is modular covariant. We show that this is the case both before and after projection, and that modular covariance properties are preserved under both exact projection and under JK projection which was recently introduced by Pu, Wu, and Jain (PRB 96, 195302 (2017)). It is crucial for the modular properties to hold that the CF state is a proper state, i.e. that there are no holes in the occupied $\mathrm{\Lambda }$-levels.