Congruence subgroups and super-modular categories Academic Article uri icon

abstract

  • © 2018 Mathematical Sciences Publishers. A super-modular category is a unitary premodular category with Müger 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, 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 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.

altmetric score

  • 1.25

author list (cited authors)

  • Bonderson, P., Rowell, E., Wang, Z., & Zhang, Q.

citation count

  • 3

complete list of authors

  • Bonderson, Parsa||Rowell, Eric||Wang, Zhenghan||Zhang, Qing

publication date

  • July 2018