Classification of Fermionic Topological Orders from Congruence Representations

TL;DR

This paper develops a method to classify fermionic topological orders by analyzing congruence representations of the modular group, successfully classifying data up to rank 10 and discovering new classes.

Contribution

It introduces a novel classification approach for super-modular categories using congruence representations, expanding known data and identifying new modular classes.

Findings

Classified super-modular category data up to rank 10

Discovered new modular data classes at rank 10

Determined central charges without explicit modular extensions

Abstract

The fusion rules and braiding statistics of anyons in $(2+1)$D fermionic topological orders are characterized by the modular data of a super-modular category. On the other hand, the modular data of a super-modular category form a congruence representation of the $\Gamma_\theta$ subgroup of the modular group $\mathrm{SL}_2(\mathbb{Z})$. We provide a method to classify the modular data of super-modular categories by first obtaining the congruence representations of $\Gamma_\theta$ and then building candidate modular data out of those representations. We carry out this classification up to rank $10$. We obtain both unitary and non-unitary modular data, including all previously known unitary modular data, and also discover new classes of modular data of rank $10$. We also determine the central charges of all these modular data, without explicitly computing their modular extensions.