Modular fusion categories with trivial Torelli group actions

Abstract

In this paper, we study modular fusion categories whose mapping class group representations are trivial on the Torelli groups, with particular emphasis on the congruence properties of the resulting representations of $\mathrm{Sp}(2g,\mathbb{Z})$. We prove that, for any $g \ge 3$, the Torelli group is contained in the kernel of the genus-$g$ mapping class group representation $\rho_g$ associated with a modular fusion category $\mathcal{C}$ if and only if $\mathcal{C}$ is pointed. This refines the corresponding result in \cite{MW25}. We also characterize the modular fusion categories for which the genus-$2$ Torelli group is contained in $\ker \rho_2$; in particular, categories with isotropic adjoint subcategories belong to this class. Finally, we give a complete classification of modular fusion categories with isotropic adjoint subcategories.