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## "Chiral topologically ordered states on a lattice from vertex operator algebras", Nikita Sopenko, 2023

*Reviewed February 10, 2024*

*Citation:* Sopenko, Nikita. "Chiral topologically ordered states on a lattice from vertex operator algebras." arXiv preprint arXiv:2301.08697 (2023).

*Web:* https://arxiv.org/abs/2301.08697

*Tags:* Vertex-operator-algebras, Quantum-hall-effect, TQFT

In this paper, it is shown that every good vertex operator algebra can be used to construct
an on-site lattice model for its corresponding topological phase. As background, it
has been known since Huang's seminal paper that the representation category
of every well-behaved VOA is modular:

> Huang, Yi-Zhi. "Vertex operator algebras, the Verlinde conjecture, and modular tensor categories." Proceedings of the National Academy of Sciences 102.15 (2005): 5352-5356.

This means that every well-behaved VOA should have an associated topological phase.
It just so happens that these phases can have nice lattice models explicitly constructed for them,
as shown in this paper. The lattice model presented is very reminiscent of the fractional quantum Hall effect.

This paper works in high generality, with infinite dimensional Hilbert spaces at each site. A more causal
treatment which only adresses the finite dimensional case can be found in Nikita Sopenko's PhD thesis:

> Sopenko, Nikita A. Topological invariants of gapped quantum lattice systems. Diss. California Institute of Technology, 2023.

Vertex operator algebras are intimately liked with conformal field theory,
and conformal field theory is intimately linked with topological quantum field theory.
This seems to be an amazing place for topological quantum field theorists to become familiar
with vertex operator algebras.