Operator algebras

A group acting outerly on a hyperfinite II1 factor can be recovered from the inclusion of its fixed point algebra.

A general subfactor encodes a more sophisticated symmetry, including but going beyond quantum groups.

These symmetries are closely related to tensor categories and have played important roles in knot theory, quantum groups, statistical mechanics in two dimensions, Hecke algebras and topological quantum field theory.

A fruitful circle of ideas has developed in the last twenty years linking the theory of (type III) subfactors with modular invariant partition functions in conformal field theory through the use of braided subfactors and alpha-induction.

Previous work has focused primarily on rational conformal field theories, where one works with a finite system of (inequivalent) endomorphisms. This non-degenerately braided system of endomorphisms is a modular tensor category, whose simple objects are the irreducible endomorphisms and with the Hom-spaces as their morphisms. The full system of endomorphisms induced by alpha-induction is not braided in general, however the quantum double construction provides a way to construct a braided subfactor.

It is natural to seek to extend these ideas to quasi-rational tensor categories, where one now considers infinite systems of endomorphisms which have finite fusion rules, that is, only finitely many irreducible endomorphisms appear in the decomposition of any product of irreducible endomorphisms.

Such categories and associated subfactors have been studied very little, with the focus being primarily on the easier rational case. To study quantum double subfactors arising from quasi-rational systems, one is forced to consider subfactors with infinite index. Very little is known about such subfactors, although they have begun to be studied again in recent years in the case of type II1 subfactors.

The aim of this project is to construct subfactors associated to quasi-rational tensor categories and investigate their properties.

We are interested in pursuing this project and welcome applications if you are self-funded or have funding from other sources, including government sponsorships or your employer.

Please contact the supervisor when you want to pursue this project, citing the project title in your email, or find out more about our PhD programme in Mathematics.

Supervisors

Photograph of Dr Mathew Pugh

Dr Mathew Pugh

Welsh Medium Lecturer

Welsh speaking
Email:
pughmj@cardiff.ac.uk
Telephone:
+44 (0)29 2087 6862

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