Year: 2026
Type: preprint
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 $ρ_g$ 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 $\ker ρ_2$; 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, and determine the groups $G$ satisfying $\mathcal{Z}(\mathrm{Vect}_G)$ is 2-Torelli trivial but $\mathcal{Z}(\mathrm{Vect}_G)_{\mathrm{ad}}$ is not isotropic are precisely 2-Engel groups of nilpotent class 3.
Source: arXiv (Cornell University)
Authors: Chang, Liang, Ng, Siu-Hung, Wang, Yilong
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G3218341711
OpenAlex ID: W7196932644