Subfactors and conformal field theory

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In this project the principal investigator will study interactions between subfactors and conformal field theories, a topic that has proved to be very fruitful lately. The main emphasis of this research is placed on two aspects of the subject: (1) the study of supersymmetric and exotic conformal field theories using operator algebraic techniques, and (2) the study of a vast generalization of a conjecture from finite group theory to the general setting of subfactors. The applications of these operator algebraic techniques are expected to produce answers to a number of representation theory questions and to shed light on certain exotic subfactors, such as those arising from conformal inclusions. The general framework of subfactors will also provide new insight into some old group theory conjectures. The theory of operator algebras was introduced by John von Neumann in order to provide a proper mathematical framework for quantum mechanics. Noncommutativity (i.e., the fact that a product AB may differ from BA), which is a key feature of quantum mechanics, is an important aspect of operator algebras. Vaughan Jones's subfactor theory is built on this noncommutative framework and has provided linkages of operator algebras with some major parts of mathematics. Conformal field theory is a theory describing critical phenomena in condensed matter physics, and it also plays an important role in string theory. The remarkable interactions between subfactors and conformal field theory have led to many interesting mathematical developments. The aim of this research is to find solutions to some of the important mathematical issues that surface in this context and that have a wide range of applications.