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**Notes on subfactors and statistical mechanics.**
*(English)*
Zbl 0725.46038

Braid group, knot theory and statistical mechanics, Adv. Ser. Math. Phys. 9, 1-25 (1989).

[For the entire collection see Zbl 0716.00010.]

A lot has been made in the last few years of connections between knot theory, statistical mechanics, field theory and von Neumann algebras. Because of their more technical nature, the von Neumann algebras have tended to be neglected in surveys. This is not an accurate reflection of their fundamental role in the subject, both as a continuing inspiration and as the vehicle of the discovery of the original ties between statistical mechanics and knot theory.

In this largely expository article, we attempt to redress this balance by talking almost entirely about von Neumann algebras and how they occur as algebras of transfer matrices in statistical mechanical models. We shall focus mostly on the Potts model and the model of Fateev and Zamolodchikov, with a brief exposition of how vertex models are related to type III factors.

A lot has been made in the last few years of connections between knot theory, statistical mechanics, field theory and von Neumann algebras. Because of their more technical nature, the von Neumann algebras have tended to be neglected in surveys. This is not an accurate reflection of their fundamental role in the subject, both as a continuing inspiration and as the vehicle of the discovery of the original ties between statistical mechanics and knot theory.

In this largely expository article, we attempt to redress this balance by talking almost entirely about von Neumann algebras and how they occur as algebras of transfer matrices in statistical mechanical models. We shall focus mostly on the Potts model and the model of Fateev and Zamolodchikov, with a brief exposition of how vertex models are related to type III factors.

### MSC:

46L35 | Classifications of \(C^*\)-algebras |

46N55 | Applications of functional analysis in statistical physics |

82C03 | Foundations of time-dependent statistical mechanics |