Classifying subfactors and fusion categories

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Symmetry plays an important role in mathematics and in the biological and physical sciences. For example, a theorem of Emmy Noether states that symmetries of physical systems, like time and space translation, correspond to conserved quantities, like energy and momentum, respectively. Von Neumann, in his study of quantum mechanics, discovered that certain operator algebras on Hilbert space describe symmetries of quantum systems. These von Neumann algebras are built from basic building blocks called factors. A subfactor is an inclusion of factors, and its representation theory encodes quantum symmetries. In the classical setting, the symmetries of a particular object form a group, like the collection of symmetries of a square or of a molecule. When one passes from the classical setting to the quantum setting, these groups are replaced by so-called quantum groups and tensor categories. Unitary tensor categories arise naturally in the study of subfactors, and in return, subfactor theory provides a wealth of techniques for classification and construction of examples. Moreover, the quantum doubles of unitary fusion categories are unitary modular categories, which are vital to research in topological phases of matter and topological quantum computation.The first aim of this project is the classification of subfactors and fusion categories. The small index subfactor classification program has seen recent success classifying up to index five, and the principal investigator will raise this index bound slightly above five. To raise the bound even further, up towards six, new techniques and obstructions are necessary. The project will also develop more techniques for studying infinite index subfactors, where there are relatively few results. The second aim is developing deeper connections between subfactors and free probability, C*-algebras, noncommutative geometry, and conformal field theory (CFT). Recent work of Guionnet, Jones, and Shlyakhtenko developed a connection between subfactors, random matrices, and free probability. With Hartglass, the principal investigator developed this connection, discovering new connections to C*-algebras and noncommutative geometry via work of Pimsner and Voiculescu. The project will continue to investigate these new developments. Finally, conformal nets on the circle intimately relate subfactors and CFT. In joint work with Henriques and Tener, the principal investigator will study conformal planar algebras, which are a common generalization of Jones's subfactor planar algebras and genus-zero Segal CFT. Tener and the principal investigator anticipate a classification in terms of module categories for the representation category of this CFT. They also conjecture the subfactor/planar algebra duality extends to a duality between conformal planar algebras and certain morphisms in the 3-category of conformal nets.