Modular fusion categories with trivial Torelli group actions

@misc{chang2026modular,
  title = {{Modular fusion categories with trivial Torelli group actions}},
  author = {Liang Chang and Siu-Hung Ng and Yilong Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.04462},
  eprint = {2608.04462},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.04462},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Modular fusion categories with trivial Torelli group actions
AU  - Liang Chang
AU  - Siu-Hung Ng
AU  - Yilong Wang
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2608.04462
UR  - https://arxiv.org/abs/2608.04462
AB  - 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 Sp(2g, ℤ). We prove that, for any g ≥ 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 𝒞 if and only if 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 𝒵(Vect_G) is 2-Torelli trivial but 𝒵(Vect_G)_ad is not isotropic are precisely 2-Engel groups of nilpotent class 3.
ER  -