| dc.description.abstract | A super-modular category is a unitary pre-modular category with Müger center equivalent to the symmetric unitary category of super-vector spaces. Physically, super-modular categories describe universal properties of fermionic topological phases of matter. Mathematically, supermodular categories are important alternatives to modular categories as any unitary pre-modular category is the equivariantization of either a modular or a super-modular category. Unlike the modular case, one does not have a representation of the modular group SL(2, Z) associated to a super-modular category, but it is possible to obtain a representation of the index 3 θ-subgroup: Γθ < SL(2, Z). 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 Γθ representation is a congruence subgroup. We prove this conjecture for any supermodular category that is a subcategory of a modular category of twice its dimension, i.e., admitting a minimal modular extension. We also study algebraic methods for classifying super-modular categories by rank. In related work, it was shown that up to fusion rules the only non-split super-modular categories of rank ≤ 6 are PSU(2)4m+2 for m ∈ {0, 1, 2}. We develop super-modular analogs of theorems and techniques previously used in the modular setting. As an application, we classify rank 8 supermodular categories up to Grothendieck equivalence with certain restrictions. In particular, we find three prime super-modular categories of rank 8. | en |
