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In Kitaev's 2006 paper Anyons in an exactly solved model and beyond, he lists nine open questions under the Section 10 "Odds and Ends". Briefly, these are

  1. Find a condensed matter realization of the Kitaev model.
  2. Use Abelian anyons for quantum memories.
  3. Realize anyons in a lattice model with spontaneous time reversal symmetry breaking.
  4. Create a stable four Majorana model in a Heisenberg antiferromagnet.
  5. Extend the $\nu \mod 16 $ classification of topological phases of $\mathbb{Z}_2$ gauge fields coupled to fermions to multilayer systems.
  6. Prove or disprove that a topological phase is characterized by the chiral central charge $c_-$ and a modular tensory category.
  7. Prove or disprove that CFTs are topological equivalent if they obey the conjecture in 6, ie there is a consistent 1D boundary between them.
  8. A comprehensive study of BEC of the anyon vortices.
  9. A mathematical framework for describing direct products of $\nu$ theories.

How many of these questions have been 'solved'?

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1 has been partially solved in the following sense: Iridates amazingly have Kitaev interaction (among other interactions).

2 a lot has been known about this kind of phenomena, now part of the story of symmetry-enriched topological phase. Generally, defects of anyon-permuting symmetry must be non-Abelian and can be used to store quantum information. Ising-like defects in toric code were worked out in https://arxiv.org/abs/1004.1838.

3 has been solved by Yao and Kivelson: https://arxiv.org/abs/0708.0040

5 if the "multilayer" part is understood as some other global symmetry, then this is a problem of what's now called symmetry enriched topological phases and a great deal has been understood.

6 is wildly open if "topological phase" means some gapped phase on lattice. If topological phase is replaced by a topological quantum field theory, then I think it is settled. In fact now the classification of TQFTs has been done in 3+1 as well (works of Lan-Kong-Wen and Johnson-Freyd).

7 there has been many progress in establishing this conjecture (within some mathematical framework for CFT). The most recent one is https://arxiv.org/abs/1905.04924 and a follow up from Liang Kong and Hao Zheng.

8 and 9: the two go under the umbrella of "anyon condensation". Assuming that topological phases are described by MTC, then there is a by now well-developed theory of what it means to form a BEC of (bosonic) anyons (a good reference is https://arxiv.org/abs/1307.8244). In addition, the "E_8" state Kitaev mentioned in 8 is now understood as an example of invertible topological phase in 2+1.

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  • $\begingroup$ Thanks for such a comprehensive answer. Are 8 and 9 not particularly interesting to the community, or just less well posed? $\endgroup$ Jul 21 at 12:55
  • $\begingroup$ @AndrewHardy I'm going to update the answer about 8 and 9. $\endgroup$
    – Meng Cheng
    Jul 21 at 12:59

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