*Citation:* Bonderson, Parsa, et al. "Congruence subgroups and super-modular categories." Pacific Journal of Mathematics 296.2 (2018): 257-270.

*Web:* https://arxiv.org/abs/1704.02041

*Tags:* Mathematical, Modular-tensor-categories, Fermionic-order

This paper gives the first serious study into the modular representations of super-modular categories. The important point is that when mapping class group actions are enacted on the torus, spin structures are permuted. The subgroup of mapping class group actions which fix a distinguished spin structure is isomorphic to a certain index-3 level-2 congruence subgroup.

The S and T matrices a supermodular category decompose into a Kronecker product of a 2x2 matrix coming from the copy of sVec and and an invertible matrix. These invertible matrices are the matrices which give the action of the mapping class group. They don't give a full SL(2,Z) action. Instead, they only give a projective representation pof the index-3 level-2 congruence subgroup reference before.

A super-modular version of the Schauenburg-Ng theorem is then given: the kernel of the modular representation is a congruence subgroup! The proof given is conditional on the existence of a minimal modular extension, which is now known to exist due to work of Johnson-Freyd.

Another takeaway from this paper is a theorem from

> Sawin, Stephen F. "Invariants of Spin three-manifolds from Chern-Simons theory and finite-dimensional Hopf algebras." Advances in Mathematics 165.1 (2002): 35-70.

which the authors highlight. Namely, the fact that every unitary ribbon fusion category is the equivariantization of a modular or super-modular tensor category. The idea is simple. Consider the Muger center of the URFC. This category is a symmetric fusion category. Deligne's theorem on tensor categories establishes Tannaka duality between symmetric tensor categories and supergroups. Being a fusion category adds a finiteness condition. This establishes a Tannaka duality between symmetric tensor categories and finite super groups, which is a pair (G,z) where G is a finite group and z is a central element of order at most 2. If z has order 1 (i.e. is the identity) then this means that category is Tannakian. De-equivariantizing this Tannakian subcategory gives a a new centerless (modular) tensor category. If z has order 2, then the representation category of (G,z) has an index-2 maximal Tannakian subcategory. De-equivariantizing this Tannakian subcategory gives a supermodular tensor category. Hence, every unitary ribbon fusion category is the equivariantization of someone modular or supermodular. A good reference for this finite version of Deligne's theorem, especially in the context of supermodular categories, is

> Bruillard, Paul, et al. "Fermionic modular categories and the 16-fold way." Journal of Mathematical Physics 58.4 (2017).