AI 中文总结
该研究明确了亏格≥3时映射类群表示核含托雷利群的模融合范畴必为点态，刻画了亏格2时满足该条件的范畴并完成了迷向伴随子范畴模融合范畴的分类，改进了已有相关结果。
AI 中文摘要
本文研究的是映射类群表示在托雷利群上平凡的模融合范畴，特别关注由此得到的$\boldsymbol{\text{Sp}(2g,\boldsymbol{\text{Z}})}$表示的同余性质。我们证明：对任意$g\boldsymbol{\text{≥}3}$，与模融合范畴$\boldsymbol{\text{C}}$相关的亏格为$g$的映射类群表示$\boldsymbol{\rho_g}$的核包含托雷利群，当且仅当$\boldsymbol{\text{C}}$是点态的，这改进了文献\boldsymbol{[\text{MW25}]}中的对应结果。我们还刻画了亏格为2的托雷利群包含于$\boldsymbol{\rho_2}$核的模融合范畴，特别地，具有迷向伴随子范畴的范畴属于此类。最后，我们对具有迷向伴随子范畴的模融合范畴给出了完整分类。
英文摘要
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.
