Congruence subgroups and super-modular categories
Congruence subgroups and super-modular categories
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DOI：
10.2140/pjm.2018.296.257
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发表时间：
2017-04
影响因子：
0.6
通讯作者：
Parsa Bonderson;E. Rowell;Qing Zhang;Zhenghan Wang
中科院分区：
文献类型：
--
作者：
Parsa Bonderson;E. Rowell;Qing Zhang;Zhenghan Wang
超模范畴是具有Muger中心的酉预模范畴，它等价于超向量空间的对称酉范畴。超模范畴是模范畴的重要替代，因为任何酉前模范畴都是模范畴或超模范畴的等变化。物理上，超模范畴描述了物质的费米拓扑相准粒子的普适性质。一般来说，没有与超模范畴相关联的模群$\mathrm{SL}（2，\mathbb {Z}）$的表示，但可以得到（下标3）$\theta$-子群的表示：$\Gamma_\theta<\mathrm{SL}（2，\mathbb {Z}）$。我们研究了这个表示的象，并猜想了一个超模的模范畴的Ng-Schauenburg同余子群定理的类似物，即$\Gamma_\theta$表示的核是一个同余子群.我们证明了这一猜想的任何超模范畴，是一个子范畴的模范畴的两倍，其维数，即允许一个最小的模扩张。猜想上，每一个超模范畴容许（正好16个）极小模扩张，因此，我们的猜想将是一个推论。
A super-modular category is a unitary pre-modular category with Muger center equivalent to the symmetric unitary category of super-vector spaces. Super-modular categories are important alternatives to modular categories as any unitary pre-modular 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 $\mathrm{SL}(2,\mathbb{Z})$ associated to a super-modular category, but it is possible to obtain a representation of the (index 3) $\theta$-subgroup: $\Gamma_\theta<\mathrm{SL}(2,\mathbb{Z})$. 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 $\Gamma_\theta$ 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, therefore, our conjecture would be a consequence.