Congruence subgroups and super-modular categories

Parsa Bonderson, Eric C. Rowell, Zhenghan Wang and Qing Zhang

Vol. 296 (2018), No. 2, 257–270

DOI: 10.2140/pjm.2018.296.257

Abstract

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 $SL\left(2,\mathbb{Z}\right)$ associated to a super-modular category, but it is possible to obtain a representation of the (index 3) $\theta$-subgroup: ${\Gamma }_{\theta }<SL\left(2,\mathbb{Z}\right)$. 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 ${\Gamma }_{\theta }$ 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.

Keywords

spin mapping class group, super-modular category, fermionic modular category

Mathematical Subject Classification 2010

Primary: 18D10

Milestones

Received: 15 April 2017

Accepted: 27 January 2018

Published: 16 July 2018