CONGRUENCE SUBGROUPS AND SUPER-MODULAR CATEGORIES
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2018 Mathematical Sciences Publishers. A super-modular category is a unitary premodular category with Mger center equivalent to the symmetric unitary category of super-vector spaces. Super-modular categories are important alternatives to modular categories as any unitary premodular category is the equivariantization of a either a modular or super-modular category. Physically, super-modular categories describe universal properties of quasiparticles in fermionic topological phases of matter. In general one does not have a representation of the modular group SL(2, ) associated to a super-modular category, but it is possible to obtain a representation of the (index 3) -subgroup: < SL(2, ). We study the image of this representation and conjecture a super-modular analogue of the Ng-Schauenburg congruence subgroup theorem for modular categories, namely that the kernel of the representation is a congruence subgroup. We prove this conjecture for any super-modular category that is a subcategory of modular category of twice its dimension, i.e., admitting a minimal modular extension. Conjecturally, every super-modular category admits (precisely 16) minimal modular extensions and our conjecture would be a consequence.
