(Fractional) quantum Hall systems are $2+1$-dimensional models which are said to possess topological order. One (maybe even complete) set of invariants of topological phases in $2+1$ dimensions is given that anyon statistics, or in mathematical terms, the (2-extended) axiomatic TQFT of the phase. Both are labelled by modular fusion categories.


**Question**: Which are the modular fusion categories associated to quantum Hall systems? Is there a table somewhere in the literature that associates to each value of the "filling fraction" (and whatever other parameters there are in quantum Hall models) a modular fusion category?


Also: Quantum Hall systems involve fermionic degrees of freedom. Are they proper fermionic phases? Or are the fermions somehow restricted to local islands of even parity, such that the resulting phases are still bosonic? In the former case, the anyon statistics are not described by modular fusion categories but their fermionic analogue, something like "modular super-fusion categories".


I guess this is written down at a lot of places, I'm just having a hard time finding it without having to skim through lots of condensed matter literature that I don't understand.



What are the topological phases of quantum Hall systems?
[Fermionic modular categories and the 16-fold way](https://arxiv.org/abs/1603.09294) classifies the topological phases of the fractional quantum Hall effect. The Laughlin states (Abelian anyons at filling factor $1/Q$, $Q$ odd) are discussed in example 2.6. The Moore-Read state (non-Abelian Ising anyons at half-integer filling factor) is in example 2.4.


